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Section 1.8: Statements of Logic

Section 1.8: Statements of Logic. Declarative vs. Conditional. Consider Theorem 1 dealing with right angles Declarative form: “Two right angles are congruent.” Conditional form: “If two angles are right angles, then they are congruent.”. Conditional Statement. 2 important parts

vernon-rowland
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Section 1.8: Statements of Logic

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  1. Section 1.8: Statements of Logic

  2. Declarative vs. Conditional • Consider Theorem 1 dealing with right angles • Declarative form: “Two right angles are congruent.” • Conditional form: “If two angles are right angles, then they are congruent.”

  3. Conditional Statement • 2 important parts • Hypothesis: The clause following if • Conclusion: The clause following then • Logic statement: “If p, then q” is written as “p=>q” (read “p implies q”) • Find the hypothesis and conclusion in the statement: • “If it is raining, then there are clouds in the sky”

  4. Negation • A negation of a statement is the opposite of a statement (~p) • Examples: • “It is sunny” becomes “It is not sunny” • “It is not cloudy” becomes “It is cloudy” • What is the negation of “It is not easy to negate statements”

  5. Converse, Inverse, and Contraposititve • Converse: switch the hypotheses and conclusion • “If p, then q” becomes “If q, then p” • Inverse: negate the hypotheses and conclusion • “If p, then q” becomes “If ~p, then ~q” • Contrapositive: switch and negate the hypotheses and conclusion • “If p, then q” becomes “If ~q, then ~p”

  6. Example • Find the converse, inverse, and contrapositive of “If you live in Chicago, then you live in Illinois” • Converse “If you live in Illinois, you live in Chicago” (not a true statement) • Inverse “If you don’t live in Chicago, then you don’t live in Illinois” (not a true statement) • Contrapositive “If you don’t live in Illinois, then you don’t live in Chicago (a true statement)

  7. Try this example • Find the converse, inverse, and contrapositive of “If you like beans, then you like vegetables.” • Converse “If you like vegetables, then you like beans.” (not always true) • Inverse “If you don’t like beans, then you don’t like vegetables.” (not always true) • Contrapositive “If you don’t like vegetables, then you don’t like beans” (true)

  8. Important Tidbits • Theorem 3: If a conditional statement is true then the contrapositive is also true • Chain of reasoning • If p=>q and q=>r, the p=>r. • Example: “If you have an Xbox, then you have a TV” and “If you have a TV, then you watch TV” becomes… “If you have an Xbox, then you watch TV”

  9. Everybody’s favorite… Homework time! 1.7: 14 1.8: 3-5, 7-10

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