Inductive ReasoningInductive reasoning is the process of using examples and observations to reach a conclusion. Any time you use a pattern to predict what will come next, you are using inductive reasoning.A conclusion based on inductive reasoning is called a conjecture.
CounterexamplesA conjecture is either true all of the time, or it is false.If we wish to demonstrate that a conjecture is true all the time, we need to prove it through deductive reasoning. We will have more on deductive reasoning and the proof process later. But for now, know that we can never prove an idea by offering examples that support the idea.However, it can be easy to demonstrate that a conjecture is false. We simply need to provide a counterexample.
Intro to Logic A statement is a sentence that is either true or false (its truth value). Logically speaking, a statement is either true or false. What are the values of these statements? The sun is hot. The moon is made of cheese. A triangle has three sides. The area of a circle is 2πr. Statements can be joined together in various ways to make new statements.
Conditional Statements A conditional (or propositional) statement has two parts: A hypothesis (or condition, or premise) A conclusion (or result) Many conditional statements are in “If… then…” form. Ex.: If it is raining outside, then I will get wet. A conditional statement is made of two separate statements; each part has a truth value. But the overall statement has a separate truth value. What are the values of the following statements? If today is Friday, then tomorrow is Saturday. If the sun explodes, then we can live on the moon. If a figure has four sides, then it is a square.
Conditional Statements Conditional statements don’t have to be “If… then…” See if you can determine the condition and conclusion in each of the following, and restate in “If… then…” form. An apple a day keeps the doctor away. What goes up must come down. All dogs go to heaven. Triangles have three sides.
Inverse The inverse of a statement is formed by negating both its premise and conclusion. Statement: IfI take out my cell phone, then Mr. Peterson will confiscate it. Inverse: If I do not take out my cell phone, then Mr. Peterson will not confiscate it.
Try these Give the inverses for the following statements. (You may wish to rewrite as “If… then…” first.) Then determine the truth value of the inverse. Barking dogs give me a headache. If lines are parallel, they will not intersect. I can use the Pythagorean Theorem on right triangles. A square is a four-sided figure.
Converse A statement’s converse will switch its hypothesis and conclusion. Statement: If I am happy, then I smile. Converse: If I am happy, then I smile .
Try these Give the converses for the following statements. Then determine the truth value of the converse. If I am a horse, then I have four legs. When I’m thirsty, I drink water. All rectangles have four right angles. If a triangle is isosceles, then two of its sides are the same.
ContrapositiveA contrapositive is a combination of a converse and an inverse. The premise and conclusion switch, and both are negated.Statement: If my alarm has gone off, then I am awake.Contrapositive: If my alarm has not gone off, not then I am not awakenot .
Try these Give the contrapositives for the following statements. Then determine its truth value. If it quacks, then it is a duck. When Superman touches kryptonite, he gets sick. If two figures are congruent, they have the same shape and size. A pentagon has five sides. Note: A contrapositive always has the same truth value as the original statement!
Symbolic representation Logic is an area of study, related to math (and computer science and other fields). In formal logic, we can represent statements symbolically (using symbols). Some common symbols: p a statement, usually a premise q a statement, usually a conclusion→ or ⇒ creates a conditional statement~ or ¬ negates a statement (takes its opposite)
Examples If p, then q p→q Inverse: If not p, then not q ~ p →~ q Converse: If q, then p q→ p Contrapositive If not q, then not p ~ q →~ p
Truth Table A truth table is a way to organize the truth values of various statements. Ina truth table, the columns are statements and the rows are possible scenarios. The table contains every possible scenario and the truth values that would occur. Example: p ~p T F F T
A conditional truth tablep q p→q T T T T F F F T T F F T
A conditional truth tablep q p→q q→p ~p →~q ~q →~p T T T T T T T F F T T F F T T F F T F F T T T T
Logical Equivalents Two statements are considered logical equivalents if they have the same truth value in all scenarios. A way to determine this is if all the values are the same in every row in a truth table.
Logical Equivalents Which of the following statements are logically equivalent?p q p→q q→p ~p →~q ~q →~p T T T T T T T F F T T F F T T F F T F F T T T T
Conjunctions A conjunction consists of two statements connected by ‘and’. Example: Water is wet and the sky is blue. Notation: A conjunction of p and q is written as p∧q
Conjunctions A conjunction is true only if both statements are true. Remember: the truthp q p ^q value of a conjunction T T T refers to the statement as a whole. T F F Consider: “The sun is F T F out and it is raining.” F F F
Disjunctions A disjunction consists of two statements connected by ‘or’. Example: I can study or I can watch TV. Notation: A disjunction of p and q is written as p∨q
Disjunctions A disjunction is true if either statement is true.p q pvq Consider: “Timmy goes to Stanton or he T T T goes to Paxon.” T F T F T T F F F
Biconditional A biconditional statement is a special type of conditional statement. It is formed by the conjunction of a statement and its converse. Example: If a quadrilateral has four right angles then it is a rectangle, and if a quadrilateral is a rectangle then it has four right angles. Biconditional statements can be shortened by using “if and only if” (iff.). A quadrilateral is a rectangle if and only if it has four right angles. This is true whether you read it forwards or ‘backwards’.
Biconditional A good definition will consist of a biconditional statement. Ex: A figure is a triangle if and only if it has three sides.
Biconditional A biconditional is true when the statements have the same truth value.p q p↔q Consider: “Two distinct coplanar lines are T T T parallel if and only if they have the same T F F slope.” F T F “Our team will win the playoffs if and only if F F T pigs fly.”
Venn Diagrams The truth values of compound statements can also be represented in Venn diagrams. p: A figure is a quadrilateral. q: A figure is convex. p q Which part of the diagram represents: p∧q p∧ ~ q p∨q ~ p∨ ~ q
Venn Diagrams – Conditionals A Venn diagram can represent a conditional statement: p: A figure is a quadrilateral. q: A figure is a square. p q