Lesson 49
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This document discusses logical arguments and conditional statements. It defines premise, conclusion, valid arguments, and conditional ("if...then") statements. It also defines and provides examples of the converse, contrapositive, and inverse of conditional statements. Specifically, it explains that the converse may be true or false, while the contrapositive of a true statement is always true.
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Dec.11 Logical Statements
1. The document discusses logical statements and their components including conditional statements with hypotheses and conclusions.
2. It provides examples of different types of logical statements including the converse, which interchanges the hypothesis and conclusion; the contrapositive, which reverses and negates both parts; and the inverse, which negates both parts but does not change their order.
3. Not all converses of true statements are necessarily true themselves.
Logic Statements:Conditional statements
Conditional statements relate a hypothesis and conclusion with an "if-then" structure. A conditional statement is false if the hypothesis is true but the conclusion is false. Some examples of conditional statements and their truth values are provided, including:
- If LJ rests for 10 days, then his ankle will heal. This would be false if LJ rested 10 days but his ankle did not heal.
- If x is prime, then x is odd. This is false because 2 is prime but not odd.
- If I study, then I will pass the exam. This would be false if I studied but still failed the exam.
Logical Operators in Brief with examples
This document defines and explains different types of logical operators: negation, conjunctions, disjunctions, conditionals, and bi-conditionals. It provides examples of each operator and includes their truth tables showing how the operators evaluate statements as true or false based on the truth values of the individual statements.
Conditional statements!
Conditional statements come in the form "If hypothesis then conclusion" and there are three other forms: the converse switches the hypothesis and conclusion, the inverse inserts "not" to both, and the contrapositive switches and inserts "not".
Conditional statements!
Conditional statements come in the form "If hypothesis then conclusion" and there are three other forms: the converse switches the hypothesis and conclusion, the inverse inserts "not" to both, and the contrapositive switches and inserts "not".
Hypothetical proposition
This document discusses different types of propositions:
1. Categorical propositions declare something unconditionally, while hypothetical propositions express a conditional relationship between two clauses.
2. There are three types of hypothetical propositions: conditional, disjunctive, and conjunctive.
3. Conditional propositions express dependence between two statements using terms like "if" and "then". Disjunctive propositions use "either/or" to present alternatives that exclude each other. Conjunctive propositions deny that two contradictory statements can both be true of the same subject.
PowerPoint Presentation
This document discusses the basics of propositional logic. It defines propositions or statements as the basic units of propositional logic. Compound propositions are formed when simple propositions are connected with logical connectives like "and" and "or". A proposition must always be able to be validated as true or false. It provides examples of true, false, and non-valid propositions. Propositional variables are used to represent unspecified statements. Logical equivalences are compound propositions that have the same logical content. Predicates are parts of statements that can be affected by variables. Quantifiers like universal, existential, and uniqueness are used to represent logical quantities.
Hypothetical & Modal Propositions
This document discusses different types of propositions: categorical propositions, hypothetical propositions, and modal propositions. It provides examples and definitions for each type. Specifically for hypothetical propositions, it defines three kinds: conditional propositions, disjunctive propositions, and conjunctive propositions. For modal propositions, it defines four modes: necessary, contingent, possible, and impossible and provides examples of each. The document concludes with exercises identifying proposition types.
Recommended
Dec.11 Logical Statements
1. The document discusses logical statements and their components including conditional statements with hypotheses and conclusions.
2. It provides examples of different types of logical statements including the converse, which interchanges the hypothesis and conclusion; the contrapositive, which reverses and negates both parts; and the inverse, which negates both parts but does not change their order.
3. Not all converses of true statements are necessarily true themselves.
Logic Statements:Conditional statements
Conditional statements relate a hypothesis and conclusion with an "if-then" structure. A conditional statement is false if the hypothesis is true but the conclusion is false. Some examples of conditional statements and their truth values are provided, including:
- If LJ rests for 10 days, then his ankle will heal. This would be false if LJ rested 10 days but his ankle did not heal.
- If x is prime, then x is odd. This is false because 2 is prime but not odd.
- If I study, then I will pass the exam. This would be false if I studied but still failed the exam.
Logical Operators in Brief with examples
This document defines and explains different types of logical operators: negation, conjunctions, disjunctions, conditionals, and bi-conditionals. It provides examples of each operator and includes their truth tables showing how the operators evaluate statements as true or false based on the truth values of the individual statements.
Conditional statements!
Conditional statements come in the form "If hypothesis then conclusion" and there are three other forms: the converse switches the hypothesis and conclusion, the inverse inserts "not" to both, and the contrapositive switches and inserts "not".
Conditional statements!
Conditional statements come in the form "If hypothesis then conclusion" and there are three other forms: the converse switches the hypothesis and conclusion, the inverse inserts "not" to both, and the contrapositive switches and inserts "not".
Hypothetical proposition
This document discusses different types of propositions:
1. Categorical propositions declare something unconditionally, while hypothetical propositions express a conditional relationship between two clauses.
2. There are three types of hypothetical propositions: conditional, disjunctive, and conjunctive.
3. Conditional propositions express dependence between two statements using terms like "if" and "then". Disjunctive propositions use "either/or" to present alternatives that exclude each other. Conjunctive propositions deny that two contradictory statements can both be true of the same subject.
PowerPoint Presentation
This document discusses the basics of propositional logic. It defines propositions or statements as the basic units of propositional logic. Compound propositions are formed when simple propositions are connected with logical connectives like "and" and "or". A proposition must always be able to be validated as true or false. It provides examples of true, false, and non-valid propositions. Propositional variables are used to represent unspecified statements. Logical equivalences are compound propositions that have the same logical content. Predicates are parts of statements that can be affected by variables. Quantifiers like universal, existential, and uniqueness are used to represent logical quantities.
Hypothetical & Modal Propositions
This document discusses different types of propositions: categorical propositions, hypothetical propositions, and modal propositions. It provides examples and definitions for each type. Specifically for hypothetical propositions, it defines three kinds: conditional propositions, disjunctive propositions, and conjunctive propositions. For modal propositions, it defines four modes: necessary, contingent, possible, and impossible and provides examples of each. The document concludes with exercises identifying proposition types.
6 sayings grammar practice
This document provides information about Latin sayings, grammar rules for the perfect passive tense, and practice conjugating verbs in the future perfect passive tense. It includes:
- Five Latin sayings and their meanings
- Explanations of the perfect passive tense structure, including that the 4th principal part is a participle and comprises two words agreeing with the subject
- Examples of transitive and intransitive verbs
- Practice conjugating the verbs laudo and ago in the future perfect passive for different subjects
Adverb clause (2)
An adverb clause is a dependent clause that modifies a verb, adjective, or other adverb. Adverb clauses begin with words like after, although, because, before, if, since, than, until, and while. There are several types of adverb clauses including concession, time, reason, contrast, result, purpose, place, and manner clauses. Adverb clauses provide details about time, place, manner, cause, condition, or comparison.
Future + time clause
This document provides information about using future time clauses in sentences. It discusses the structure of sentences with future time clauses, which have a main clause and a time clause. The main clause often uses future tense verbs like "will" or "be going to" while the time clause uses present tense verbs. Time clauses can come at the beginning or end of sentences and introduce the first or second event. The document provides examples of sentences with future time clauses and exercises for learners to practice forming sentences with future time clauses using time words like "before," "until," and "after."
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Sentence fragments
The document defines and provides examples of sentence fragments, independent clauses, dependent clauses, and phrases. It explains that fragments lack essential elements of a complete sentence, such as a subject or verb. Various ways of identifying and revising fragments are discussed, including attaching phrases or dependent clauses to nearby independent clauses.
Presentation 3
This document provides information about future time clauses in English grammar. It discusses the structure of sentences with future time clauses, which have a main clause and a time clause. The main clause often uses future tense verbs like "will" or "be going to" while the time clause uses present tense verbs. Time clauses can come at the beginning or end of sentences and are separated by commas when they begin sentences. Several examples of future time clauses are given along with exercises for learners to practice forming sentences with time clauses using time words like "when", "before", and "after".
Future time clauses
Free English Lesson
https://curious.com/jsaenz/simple-past-statements-with-be/in/intermediate-english-for-esl-students?ref=Q86D15FCP30
Hi , My name is Joe, I'm a native English teacher from the United States. Nowadays English is an enabling tool that will help you compete globally. If you would like to become fluent in this language; I can help you reach your language goals via private one on one English lessons on the internet.
I'm a respected Online English Teacher. I have been teaching English for 10 years and online since 2012
If you are interested in receiving one on one private English lessons with me ( The first class is absolutely free ) : Fill out this form with your information so that I can contact you:
Link To Form: https://docs.google.com/forms/d/1eLr-YQwPV9TAS-WuZIVJZ3ytHsHoOEHOOi2EDnnOK5g/viewform
The sentence, the phrase and the clause
This document defines and provides examples of sentences, phrases, clauses, comma splices, dependent clauses, subordinating conjunctions, dangling modifiers, run-on sentences, and mixed construction. It notes that a sentence contains both a subject and predicate, while a phrase lacks one or both. A clause also contains both subject and predicate. Improper use of punctuation like commas can create errors, and dependent and independent clauses must be properly connected.
Presentation 3 Time Clauses Referring to the Future
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Logical Connectives (w/ ampersands and arrows)
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Chapter 2 the accusative case
The document discusses the accusative case in Latin. It notes that the accusative case is used for direct objects of verbs. Direct objects receive the action of the verb and answer who or what after the verb. Verbs that can take direct objects are action verbs, not linking verbs. All singular direct objects in Latin end in -m and all plural direct objects end in -s, showing the accusative case ending. The typical word order in a Latin sentence is subject, direct object, verb. However, the case is determined by the ending, not the position.
Four kinds of clauses
This document defines and provides examples of different types of clauses:
Independent clauses can stand alone as complete sentences. Adverb clauses provide more information about the main clause and usually begin with subordinating conjunctions like "before" or "because". Adjective clauses modify nouns and usually begin with relative pronouns like "who" or "that". Noun clauses can serve as the subject, object, or object of a preposition in a sentence and often begin with "that" or a question word. The same words like "when" can be used in different types of clauses depending on their function in the sentence.
Verbs chapter 2
The document discusses Latin verb conjugation and principal parts. It notes that verbs have three principal parts: the first person singular present tense form, the infinitive, and the first person singular perfect tense form. The infinitive ending indicates the conjugation and provides the present stem. To form other verb forms, personal endings are added to the present stem. Subject-verb agreement must match in number, so singular subjects take singular verb forms and plural subjects take plural verb forms.
CONJUCTIVE ADVERBS
Conjunctive adverbs (which are sometimes also called
sentence connectors or transitional words) are commonly
used in serious business, technical, and academic writing.
Conjunctive adverbs are.similar in meaning to other types
of connecting words such as and, but, or, etc., but they
have one very important difference: they join sentences,
not parts of sentences.
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Presentation 3
This document provides information about future time clauses in English grammar. It discusses the structure of sentences with future time clauses, which have a main clause and a time clause. The main clause often uses future tense verbs like "will" or "be going to" while the time clause uses present tense verbs. Time clauses can come at the beginning or end of sentences and are separated by commas when they begin sentences. Several examples of future time clauses are given along with exercises for learners to practice forming sentences with time clauses using time words like "when", "before", and "after".
Future time clauses
Free English Lesson
https://curious.com/jsaenz/simple-past-statements-with-be/in/intermediate-english-for-esl-students?ref=Q86D15FCP30
Hi , My name is Joe, I'm a native English teacher from the United States. Nowadays English is an enabling tool that will help you compete globally. If you would like to become fluent in this language; I can help you reach your language goals via private one on one English lessons on the internet.
I'm a respected Online English Teacher. I have been teaching English for 10 years and online since 2012
If you are interested in receiving one on one private English lessons with me ( The first class is absolutely free ) : Fill out this form with your information so that I can contact you:
Link To Form: https://docs.google.com/forms/d/1eLr-YQwPV9TAS-WuZIVJZ3ytHsHoOEHOOi2EDnnOK5g/viewform
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This document defines and provides examples of sentences, phrases, clauses, comma splices, dependent clauses, subordinating conjunctions, dangling modifiers, run-on sentences, and mixed construction. It notes that a sentence contains both a subject and predicate, while a phrase lacks one or both. A clause also contains both subject and predicate. Improper use of punctuation like commas can create errors, and dependent and independent clauses must be properly connected.
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Independent clauses can stand alone as complete sentences. Adverb clauses provide more information about the main clause and usually begin with subordinating conjunctions like "before" or "because". Adjective clauses modify nouns and usually begin with relative pronouns like "who" or "that". Noun clauses can serve as the subject, object, or object of a preposition in a sentence and often begin with "that" or a question word. The same words like "when" can be used in different types of clauses depending on their function in the sentence.
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The document discusses Latin verb conjugation and principal parts. It notes that verbs have three principal parts: the first person singular present tense form, the infinitive, and the first person singular perfect tense form. The infinitive ending indicates the conjugation and provides the present stem. To form other verb forms, personal endings are added to the present stem. Subject-verb agreement must match in number, so singular subjects take singular verb forms and plural subjects take plural verb forms.
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Grade 8-if-then-statement
Here are the converses, inverses, and contrapositives of the given statements:
1. Conditional: If two angles form a linear pair, then they are supplementary.
Converse: If two angles are supplementary, then they form a linear pair.
Inverse: If two angles do not form a linear pair, then they are not supplementary.
Contrapositive: If two angles are not supplementary, then they do not form a linear pair.
2. Conditional: If a parallelogram has a right angle, then it is a rectangle.
Converse: If a parallelogram is a rectangle, then it has a right angle.
Inverse: If a parallelogram does not have
G8 Math Q2- Week 8- Transforming Contrapositive, Converse and Inverse.pptx
This document discusses conditional statements and their transformations. It defines the hypothesis and conclusion of an if-then statement and shows examples of converting a statement to its converse, inverse, and contrapositive. Specifically, it demonstrates that the converse switches the hypothesis and conclusion, the inverse negates both, and the contrapositive switches and negates both. The document provides examples, activities for students to practice the transformations, and emphasizes the importance of identifying the hypothesis and conclusion of an if-then statement.
Geometry conditional statements and their converse
The document discusses writing conditional statements in if-then form and their converses. It defines key terms like conditional statement, hypothesis, and conclusion. It explains that conditional statements have two parts - a hypothesis and conclusion - and discusses how to identify these parts and determine if a conditional statement is true or not. It also covers writing the converse of a conditional statement by exchanging the hypothesis and conclusion.
Inductive reasoning & logic
Inductive reasoning uses examples and observations to reach a conclusion, called a conjecture. A conjecture is either always true or false. While examples can support a conjecture, they cannot prove it. A counterexample can demonstrate that a conjecture is false.
Geometry unit 2.2
Conditional statements relate two parts, a hypothesis and conclusion, using "if-then" logic. The hypothesis follows "if" and is the given condition, while the conclusion follows "then" and is the result. Symbols like implies (®), not (~), and (Ù,Ú) can also represent conditional statements and connect multiple statements. A conditional statement is only false if the hypothesis is true but conclusion is false. Contrapositives and converses of conditional statements are also considered in determining logical equivalencies.
Inductive and Deductive Reasoning
The document discusses inductive and deductive reasoning. Inductive reasoning involves forming general conclusions from specific observations, while deductive reasoning draws specific conclusions from general statements. Examples are given of inductive arguments building from specific cases to a general rule, and deductive arguments applying a general premise to specific cases. The key features of deductive reasoning, including conditional statements and the five types of if-then logical structures (conditional, converse, counter example, inverse, and contrapositive), are also explained through examples.
IF_THEN_STATEMENT.pptx
The document defines conditional statements and identifies their key components. A conditional statement consists of two propositions: a hypothesis (p) and a conclusion (q) connected by "if-then". The hypothesis states a condition that if true, requires the conclusion to also be true. Examples of conditional statements are provided and their hypotheses and conclusions are identified. The relationship between the hypothesis and conclusion determines if a conditional statement is true or false.
Geom 2point1
This document discusses conditional statements and their components. It defines a conditional statement as having a hypothesis and conclusion connected by "if...then". It provides examples of writing the converse, inverse, and contrapositive of conditional statements. The key points are:
- A conditional statement has two parts: a hypothesis and conclusion connected by "if...then"
- The converse switches the hypothesis and conclusion
- The inverse negates both the hypothesis and conclusion
- A statement is equivalent to its contrapositive
1.3.1 Conditional Statements
* Identify, write, and analyze conditional statements
* Write the inverse, converse, and contrapositive of a conditional statement
* Write a counterexample to a false conjecture
* Write a biconditional statement
1.3.2 Conditional Statements
* Identify, write, and analyze the truth value of conditional statements.
* Write the inverse, converse, and contrapositive of a conditional statement.
1.4 Conditional Statements
* Identify, write, and analyze conditional statements
* Write the inverse, converse, and contrapositive of a conditional statement
* Write a counterexample to a fake conjecture
Conditional Statements
The document discusses conditional statements and their components. It provides examples of identifying the hypothesis and conclusion of conditional statements. It also discusses determining the truth value of conditional statements based on whether the hypothesis and conclusion are both true, false, or one is true and one is false. Examples are given for writing the converse, inverse, and contrapositive of conditional statements and determining if they are true or false, providing counterexamples when false.
Conditional Statements
The document discusses conditional statements and their components. It provides examples of identifying the hypothesis and conclusion of conditional statements. It also discusses determining the truth value of conditional statements based on whether the hypothesis and conclusion are both true, false, or one is true and one is false. Finally, it discusses writing the converse, inverse, and contrapositive of conditional statements and determining if those related statements are true or false, providing counterexamples when they are false.
Conditional Statements
The document discusses conditional statements and their components. It provides examples of identifying the hypothesis and conclusion of conditional statements. It also discusses determining the truth value of conditional statements based on whether the hypothesis and conclusion are both true, false, or one is true and one is false. Finally, it discusses writing the converse, inverse, and contrapositive of conditional statements and determining if those related statements are true or false, providing counterexamples when they are false.
Conditional Statements
The document discusses conditional statements and their components. It provides examples of identifying the hypothesis and conclusion of conditional statements. It also discusses determining the truth value of conditional statements based on whether the hypothesis and conclusion are both true, false, or one is true and one is false. Examples are given using conditional statements about polygons and Yukon's ankle healing. The relationships between conditionals, converses, inverses and contrapositives are also explained.
If and then statements
The document discusses conditional statements, also known as if-then statements. It defines conditional statements as having two parts: a hypothesis and a conclusion. The hypothesis is the if part and the conclusion is the then part. It provides examples of writing conditional statements based on given inputs and outputs. It also discusses determining the converse, inverse, and contrapositive of a conditional statement by changing or negating the hypothesis and conclusion.
Geometry Section 2-2
This document defines key vocabulary terms related to statements, conditionals, and biconditionals. It defines statements, truth values, negation, and compound statements. It then defines conjunctions, disjunctions, if-then statements, hypotheses, conclusions, related conditionals, converses, inverses, contrapositives, and biconditionals. It provides examples of writing compound statements and determining their truth values. It also provides examples of identifying hypotheses and conclusions of conditional statements and writing them in if-then form. The document aims to build foundational understanding of propositional logic.
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More from Kelly Scallion
May 2011 schedule
The document contains a schedule for May and June 2011, listing exercises, tests, and exams for different days of the week. It includes Exercise #1-35 on May 11th, a test on May 12th, no school on May 13th for prom, and Victoria Day on May 23rd with no school. The June schedule focuses on provincial exam review from June 6th to 10th, with the provincial exam taking place from 8:45-12:00 on June 16th in the library.
Exercise #23
Calculators can evaluate base 10 logarithms. The change of base theorem, logb a = logc a / logc b, allows evaluating logarithms with non-base 10 bases by relating them to base 10 logarithms. Some useful properties of logarithms are that logb b = 1, logb b^y = y, and b^logb a = a. Students are assigned exercises 23 problems 1-13, 15-16, 18, and 20 to practice evaluating logarithms.
Exercise #21
The document discusses logarithmic functions and their inverses, exponential functions. It provides examples of logarithmic and exponential forms and how to convert between them. It also covers graphing logarithmic functions, their domains and ranges, characteristics like asymptotes. It provides examples of evaluating logarithmic expressions and transformations of logarithmic functions.
Exercise #21
This document discusses logarithmic functions and their inverses, exponential functions. It provides examples of logarithmic and exponential forms and how to convert between them. It also covers graphing logarithmic functions, their domains and ranges, characteristics like asymptotes. It provides examples of evaluating logarithmic expressions and transformations of logarithmic functions.
Exercise #22
This document outlines three theorems for logarithms:
1) The Product Theorem states that the log of a product is equal to the sum of the logs of the factors.
2) The Quotient Theorem states that the log of a quotient is equal to the difference of the logs of the factors.
3) The Power Theorem states that the log of a factor to a given power is equal to the power times the log of the factor.
These theorems only apply when the logarithms have the same base. Examples are provided to demonstrate applying the theorems to simplify logarithmic expressions and evaluate logarithms.
Exercise #19
The document discusses exponential functions of the form f(x) = a^x where a is a constant. It provides examples of exponential functions with a = 2 and a = 1/2. It describes the domain, range, and common point of exponential functions. The document also discusses transformations of exponential functions by adding or subtracting constants, and provides examples of sketching and describing transformed exponential functions. Finally, it lists common exponential expressions.
Exercise #16 notes
This document presents sum and difference identities for trigonometric functions sine and cosine. It gives the formulas for sin(α + β), cos(α + β), sin(α - β) and cos(α - β) in terms of sinα, cosα, sinβ and cosβ. It then provides examples of using these formulas to evaluate trigonometric functions at values that are not covered by special triangles.
Exercise #18 notes
This document discusses double angle identities for sine, cosine, and tangent. It provides the formulas for sin(2θ), cos(2θ), and tan(2θ) in terms of sin(θ) and cos(θ). The document also includes examples of using the double angle identities to evaluate trigonometric functions and solve equations.
Exercise #17 notes
This document contains examples of using sum and difference identities to express trigonometric functions with combined arguments in terms of single variables. It provides the identity cos(A + B) = cos(A)cos(B) - sin(A)sin(B) and uses it to express cos(π/2 + x) in terms of x alone. It also gives examples of using identities to prove relationships between trigonometric functions with combined arguments.
Exercise #13 notes ~ graphing
This document provides instructions for graphing trigonometric transformations in 3 steps: 1) Determine the a, b, c, and d values from the function's factored form. 2) Draw the median position and amplitude. 3) Determine the period and mark points to graph the wave-like function. Examples graph y=3sin(2x)-1, f(x)=sin(1/2x+1), and f(x)=2cos(3x)-2.
Exercise #11 notes
This document discusses graphing absolute value functions. It provides examples of graphing various absolute value functions, including f(x)=|x|^2, f(x)=2|x|-1, f(x)=|x|^2-1, f(x)=3|x|^2-4, f(x)=|cos(x)|, f(x)=2|x|-3-1, and shows how to write a piecewise function definition for f(x)=|x|^2. The graphs are V-shaped and symmetric about the y-axis, with vertices at the points where the absolute value terms are equal to zero.
Exercise #13 notes ~ equations
This document provides information on writing trigonometric equations from sinusoidal graphs. It outlines the two basic sinusoidal equations as f(x) = a sin(b(x - c)) + d or f(x) = a cos(b(x - c)) + d. The variables a, b, c, and d represent the amplitude, period, start point, and median, respectively. Formulas are given to identify each variable based on properties of the graph like maximum, minimum, and period.
Exercise #15 notes
This document discusses proving trigonometric identities. It lists four common operations for proving identities: 1) adding or subtracting rational expressions, 2) multiplying or dividing rational expressions, 3) factoring, and 4) multiplying by the conjugate. It then provides four examples of identities to prove using these techniques.
Exercise #14 notes
The document discusses trigonometric identities, including 8 fundamental identities. It provides examples of using identities to simplify trigonometric functions and proves identities by showing the left and right hand sides are the same. Examples include simplifying tan θ cos θ, expressing tan θ in terms of sin θ, and proving cot θ sin θ = cosθ.
Exercise #8 notes
This document discusses how to sketch graphs resulting from transformations of basic parent functions, including horizontal and vertical stretches as well as translations. It provides examples of stretching and translating graphs of f(x) = x^2 and generalizes the effects of stretches and translations on basic parent functions. The document concludes with instructions on combining multiple transformations and an example problem determining the x-intercepts resulting from a composite transformation.
Exercise #9 notes
1) Reflections of functions f(x) over the x-axis or y-axis produce new equations by changing the sign of the y-values or x-values respectively.
2) The inverse of a function f(x) is denoted f^-1(x) and is found by switching the x and y values when graphed, effectively reflecting over the line y=x.
3) For example, the inverse of f(x)=2x+5 is found by graphing f^-1(x) which treats x as the output and y as the input.
Exercise #10 notes
This document provides examples for graphing reciprocals of functions. It explains that for reciprocals, smaller numbers on the x-axis become larger on the y-axis, and vice versa. An example graphs the reciprocal of f(x)=1/x^2. It notes that reciprocals can be written in the form f(x)=k/x+h, where k and h are determined from the original function. Further examples graph the reciprocals of f(x)=2/x+4 and f(x)=sin(x).
Exercise #7 notes
This document discusses transformations of graphs including vertical and horizontal shifts. It provides examples of parent functions being shifted vertically by adding or subtracting a number, and horizontally by adding or subtracting inside or outside the function. Examples are given of shifting the graph of f(x)=sinx by 1 unit vertically and horizontally shifting and vertically shifting the graph of f(x)=x^2-3.
Statement and demand assignment
This document provides information for four tax situations including land and building assessments, mill rates, property frontages, improvement taxes, school and education levy taxes, tax credits, and penalties. The questions involve calculating statements and demands for residential properties with varying assessments, mill rates, improvement taxes, and school division taxes.
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みなさんこんにちはこれ何文字まで入るの？40文字以下不可とか本当に意味わからないけどこれ限界文字数書いてないからマジでやばい文字数いけるんじゃないの？えこ...
ここ3000字までしか入らないけどタイトルの方がたくさん文字入ると思います。
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Lesson 49
- 1. Lesson 49 Converse, Contra Positives, If…Then Statements Text: Chapter 6, section 5 A logical argument consists of a set of premises and conclusions. PREMISE is the hypothesis and the statement to be arrived at is the CONCLUSION. An argument is valid if the conclusion has been arrived at through accepted forms of reasoning. The “If…then” statement is called a Conditional Statement and is made up if the hypothesis (if) and the conclusion (then). If I do all my assignments then I will have a better pre-calculus mark. A statement may be TRUE or FALSE. Example: If two angles are vertical angles, then they are congruent. (True statement) Symbolism: p q Converse: A statement formed by interchanging the hypothesis and conclusion. Example: If two angles are congruent, then they are vertical angles (false statement) Symbolism: q p The converse statement may or may not be true. “If and only if” is used when the converse of a true statement is true. Example: A triangle has two equal sides if and only if, it has two equal angles.
- 2. Symbolism: p q If the statement is true to create the CONTRA-POSITIVE, you reverse the hypothesis and the conclusion and negate (turn to a negative word) the meaning of both hypothesis and conclusion. Example: If two angles are not congruent then they are not vertical angles. (True statement) Symbolism: q p The contra-positive of a true statement is true and the contra-positive of a false statement is still false. To create the Inverse, you negate the hypothesis and the conclusion. Example: If the weather is good I will go golfing p q Inverse – If the weather is NOT good I will NOT go golfing. p q Example: If Sunny lives in Plum Coulee, then Sunny lives in Manitoba. Write a Converse, Inverse and Contra-positive statements: Converse – Inverse – Contra-Positive