1. Conditional Statements
Lesson 2-1
Conditional
Statements

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

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

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

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

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

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

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
Statement:
If you live in Virginia, then you live in Richmond.
Is there a counterexample?
I live in Virginia, BUT I live in Glen Allen.
YES...
Therefore () the statement is false.

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

11. pq
is used to represent
if p, then q or p implies q

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

13. is used to represent the word~
is used to represent the word
“not”

14. 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

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

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

17. Example pq: A number is even and it is divisible by 3.
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...

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

19. Example pq: A number is even or it is divisible by 3.
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,...

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

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

22. Different Forms of Conditional Statements

23. Converse: q  p
pq If two angles are vertical, then they are congruent.
qp If two angles are congruent, then they are vertical.

24. Inverse: ~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.

25. Contrapositive:~q~p
pq If two angles are vertical, then they are congruent.
~q~p If two angles are not congruent, then they are not vertical.

26. If pq is true, If pq is false,
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.

27. Biconditional
When a conditional statement and its converse are both true, the two statements may be combined.
Use the phrase if and only if (iff)

28. 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.