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