Lesson 2 1 Conditional Statements Conditional Statements have

Lesson 2 -1 Conditional Statements

Conditional Statements have two parts: 4 Hypothesis (denoted by p) and 4 Conclusion (denoted by q)

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

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

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

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

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

Conditional Statements can be true or false: 4 A conditional statement is false only when the hypothesis is true, but the conclusion is false. 4 A counterexample is an example used to show that a statement is not always true and therefore false.

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

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

p q is used to represent 4 if p, then q or p implies q

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.

~ is used to represent the word 4 “not”

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

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)

is used to represent the word 4 “and”

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. . .

is used to represent the word 4 “or”

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, . . .

is used to represent the word 4 “therefore”

Example Therefore, the statement is false

Different Forms of Conditional Statements

Converse: q p 4 p q If two angles are vertical, then they are congruent. 4 q p If two angles are congruent, then they are vertical.

Inverse: ~p ~q 4 p q If two angles are vertical, then they are congruent. 4~p ~q If two angles are not vertical, then they are not congruent.

Contrapositive: ~q ~p 4 p q If two angles are vertical, then they are congruent. 4~q ~p If two angles are not congruent, then they are not vertical.

Contrapositives are logically equivalent to the original conditional statement. 4 If p q is true, then q p is true. 4 If p q is false, then q p is false.

Biconditional 4 When a conditional statement and its converse are both true, the two statements may be combined. 4 Use the phrase if and only if (iff)

Definitions are always biconditional If an angle is right then it has a measure of 90. 4 Converse: If an angle measures 90 , then it is a right angle. 4 Biconditional: An angle is right iff it measures 90. 4 Statement: