1. 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.

2. 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.

3. 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.

4. Inverse The inverse of a statement is formed by negating both its hypothesis and conclusion. Statement: If I take out my cell phone, then Mr. Peterson will confiscate it. Inverse: If I do take out my cell phone, then Mr. Peterson will confiscate it. not not

5. 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.

6. Converse A statement’s converse will switch its hypothesis and conclusion. Statement: If I am happy, then I smile. Converse: If, then. I smileI am happy

7. 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.

8. 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, then. I am not awake my alarm has not gone off not not

9. 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!

10. 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: a statement, usually a premise a statement, usually a conclusion creates a conditional statement negates a statement (takes its opposite)

11. Examples If p, then q Inverse: If not p, then not q Converse: If q, then p Contrapositive If not q, then not p

12. Truth Table A truth table is a way to organize the truth values of various statements. In a 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 TF F

13. A conditional truth table pqp → q TT TF FT FF T F T T

14. A conditional truth table pqp → q TT TF FT FF T F T T q → p~p →~q~q →~p T T F T T T F T T F T T

15. 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

16. Logical Equivalents Which of the following statements are logically equivalent? pqp → q TT TF FT FF T F T T q → p~p →~q~q →~p T T F T T T F T T F T T

17. 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

18. Conjunctions A conjunction is true only if both statements are true. pq p ^ q TT T T T F F FF F F F Remember: the truth value of a conjunction refers to the statement as a whole. Consider: “The sun is out and it is raining.”

19. 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

20. Disjunctions A disjunction is true if either statement is true. pqp v q TT T T T F F FF T T F Consider: “Timmy goes to Stanton or he goes to Paxon.”

21. Homework Page 129 #1-14 all