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Logic

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 2 π r .

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Logic

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  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 am happy I smile

  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 . my alarm has not gone off not I am not awake 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 • ContrapositiveIf 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: T F F T

  13. A conditional truth table T T T T F F F T T F F T

  14. A conditional truth table T T T T T T T F F T F T F T F F T T F F T T T T

  15. 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.

  16. Logical Equivalents • Which of the following statements are logically equivalent? T T T T T T T F F T F T F T F F T T F F T T 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. • Remember: the truth value of a conjunction refers to the statement as a whole. • Consider: “The sun is out and it is raining.” T T T T F F F T F F F F

  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. • Consider: “Timmy goes to Stanton or he goes to Paxon.” T T T T F T F T T F F F

  21. Homework • Page 129 #1-14 all

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