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# Inductive reasoning & logic

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### Inductive reasoning & logic

1. 1. Inductive ReasoningInductive 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.
2. 2. CounterexamplesA 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.
3. 3. 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.
4. 4. 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.
5. 5. 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.
6. 6. 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.
7. 7. 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.
8. 8. 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 .
9. 9. 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.
10. 10. ContrapositiveA 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 .
11. 11. 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!
12. 12. 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)
13. 13. 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
14. 14. 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
15. 15. A conditional truth tablep q p→q T T T T F F F T T F F T
16. 16. 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
17. 17. 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.
18. 18. 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
19. 19. 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
20. 20. 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
21. 21. 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
22. 22. 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
23. 23. 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’.
24. 24. Biconditional A good definition will consist of a biconditional statement. Ex: A figure is a triangle if and only if it has three sides.
25. 25. 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.”
26. 26. 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
27. 27. 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