1. Conditional Statements Lesson 2-1

2. Conditional Statements have two parts: Hypothesis ( denoted by p) and Conclusion ( denoted by q)

3. Hypothesis (p) Phrase following “if” the given information

4. Conclusion (q) Phrase following “then” the result of the given information

5. Conditional statements can be put into an “if-then” form to clarify which part is the hypothesis and which is the conclusion.

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

7. If two angles are vertical, then they are congruent. p implies q Hypothesis (p): two angles are vertical Conclusion (q): they are congruent

8. 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.

9. Giving a Counterexample Therefore (  ) the statement is false. Statement: If you live in Virginia, then you live in Richmond. True or False? Give a counterexample: Henry lives in Virginia, BUT he lives in Ashland.

10. Symbolic Logic

11. Symbols can be used to modify or connect statements.

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

13. Example H : I watch football HH Therefore, I watch football

14.  is used to represent implies used in if … then

15. 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.

16. ~ is used to represent the word “not” or “negate”

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

18. Example r: I am not happy ~r: I am happy Notice: ~r took the “not” out… it would have been a double negative (not not)

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

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

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

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

23. iff is used to represent the phrase “if and only if”

24. Example h: I watch football k: the Eagles are playing h iff k I watch football if and only if the Eagles are playing

25. Different Forms of Conditional Statements

26. A conditional statement can be written in three different ways. These three new conditional statements can be true or false. EXAMPLE: p  q If two angles are vertical, then they are congruent.

27. Converse: q  p SWITCH (p and q but not if or then) If two angles are congruent, then they are vertical.

28. Inverse: ~p  ~q NEGATION (keep same order) If two angles are not vertical, then they are not congruent.

29. Contrapositive:~ q  ~p SWITCH and NEGATE If two angles are not congruent, then they are not vertical.

30. 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.

31. Biconditional If a conditional statement and its converse are both true, then the two statements may be combined. using the phrase if and only if (iff)

32. Definitions are always biconditional Statement: If an angle is right then it has a measure of 90 . Converse: If an angle measures 90 , then it is a right angle. Biconditional: An angle is right iff it measures 90 