1. Lesson 2-1 Conditional Statements

2. Conditional Statement Defn. A conditional statement is a statement that can be written as an if- then statement. That is, as “If _____________, then ______________.”

3. Example: If your feet smell and your nose runs, then you're built upside down.

4. Conditional Statements have two parts: The hypothesis is the part of a conditional statement that follows “if” (when written in if- then form.) It is the given information, or the condition. If a number is prime, then a number has exactly two divisors. Hypothesis: a number is prime Leave off “if” and comma.

5. Conditional Statements have two parts: The conclusion is the part of a conditional statement that follows “then” (when written in if-then form.) It is the result of the given information. If a number is prime, then a number has exactly two divisors. Conclusion: a number has exactly two divisors Leave off “then” and period

6. Conditional statements can be put into an “if-then” form to clarify which part is the hypothesis and which is the conclusion. Method: Turn the subject into a hypothesis. Rewriting Conditional Statements

7. Example 1: Vertical angles are congruent. If two angles are vertical, then they are congruent. can be written as...

8. Example 2: Seals swim. If an animal is a seal, then it swims. can be written as...

9. Example 3: Babies are illogical. If a person is a baby, then the person is illogical. can be written as...

10. IF …THEN vs. IMPLIES Two angles are vertical implies they are congruent. Another way of writing an if-then statement is using the word implies.

11. Conditional Statements can be true or false: A conditional statement is false only when the hypothesis is true, but the conclusion is false. A counterexample is an example used to show that a statement is not always true and therefore false.

12. Counterexample Therefore (  ) the statement is false. Statement: If you live in Virginia, then you live in Richmond, VA. Is there a counterexample? YES... Anyone who lives in Virginia, but not Richmond, VA.

13. Symbols can be used to modify or connect statements. Symbolic Logic

14. Symbols for Hypothesis and Conclusion if p, then q or p implies q Lower case letters, such as p and q, are frequently used to represent the hypothesis and conclusion.

15. Symbols for Hypothesis and Conclusion if p, then q or p implies q Example p: a number is prime q: a number has exactly two divisors If a number is prime, then it has exactly two divisors.

16. is used to represent the words “if … then” or “implies” 

17. p  q if p, then q or p implies q means

18. Example p: a number is prime q: a number has exactly two divisors p  q: If a number is prime, then it has exactly two divisors.

19. is used to represent the word “not” ~ ~ p is the negation of p. The negation of a statement is the denial of the statement. Add or remove the word “not.” To negate, write ~ p.

20. Example p: the angle is obtuse ~p: the angle is not obtuse Be careful because ~p means that the angle could be acute, right, or straight.

21. Example p: James doesn’t like fish. ~p: James likes fish. Notice: ~p took the “not” out… it would have been a double negative (not not)

22. is used to represent the word “and” 

23. Example p: a number is even q: a number is divisible by 3 p  q: A number is even and it is divisible by 3. 6,12,18,24,30,36,42...

24. is used to represent the word “or” 

25. Example p: a number is even q: a number is divisible by 3 p  q: A number is even or it is divisible by 3. 2,3,4,6,8,9,10,12,14,15,...

26. is used to represent the word “therefore” 

27. Example Therefore, the statement is false.  the statement is false

28. Different Forms of Conditional Statements

29. Forms of Conditional Statements Converse: Statement formed from a conditional statement by switching the hypothesis and conclusion (q  p) p  q If two angles are vertical, then they are congruent. q  p If two angles are congruent, then they are vertical. Continued….. Are these statements true or false?

30. Forms of Conditional Statements Inverse: Statement formed from a conditional statement by negating both the hypothesis and conclusion. (~p  ~q) p  q : If two angles are vertical, then they are congruent. ~p  ~q: If two angles are not vertical, then they are not congruent. Are these statements true or false?

31. Forms of Conditional Statements Contrapositive: Statement formed from a conditional statement by switching and negating both the hypothesis and conclusion. (~q  ~p) p  q : If two angles are vertical, then they are congruent. ~q  ~p: If they are not congruent, then two angles are not vertical Are these statements true or false?

32. Contrapositives are logically equivalent to the original conditional statement. If p  q is true, then  q  p is true. If p  q is false, then  q  p is false.

33. Biconditional  When a conditional statement and its converse are both true, the two statements may be combined. A statement combining a conditional statement and its converse is a biconditional. Use the phrase if and only if which is abbreviated iff Use the symbol 

34. Definitions are always biconditional Statement: p  q If an angle is right then it measures 90 . Converse: q  p If an angle measures 90 , then it is right. Biconditional: p  q An angle is right iff it measures 90 .

35. Biconditional  A biconditional is in the form: Hypothesis if and only if Conclusion. or Hypothesis iff Conclusion or Hypothesis  Conclusion

36. Biconditionals in symbols Since p  q means p  q AND q  p, p  q Is equivalent to (p  q)  ( q  p)