Conditional Statements

Lesson 2-1

Conditional Statements have two

parts:

•Hypothesis (denoted by p) and

•Conclusion (denoted by q)

Hypothesis (p)

•Phrase following “if”

•the given information

Conclusion (q)

•Phrase following “then”

•the result of the given information

Conditional statements can be

put into an “if-then” form to clarify which part is the hypothesis and which is the

conclusion.

Example:Vertical angles are congruent.

If two angles are vertical, then they are congruent.

can be written as...

If two angles are vertical,

then they are congruent.

p implies q

Hypothesis (p):two angles are vertical

Conclusion (q):they are congruent

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.

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.

Symbolic Logic

Symbols can be used to modify or connect

statements.

is used to represent the word

• “therefore”

Example

H : I watch football

H

Therefore, I watch football

is used to represent

• implies • used in if … then

Example

p: a number is prime

q: a number has exactly two divisors

pq: If a number is prime, then it has exactly two divisors.

~ is used to represent the word

• “not” or

• “negate”

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

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)

is used to represent the word

• “and”

Example

a: a number is even

b: a number is divisible by 3

ab: A number is even and it is divisible by 3.

6,12,18,24,30,36,42...

is used to represent the word

• “or”

Example

a: a number is even

b: a number is divisible by 3

ab: A number is even or it is divisible by 3.

2,3,4,6,8,9,10,12,14,15,...

iff is used to represent the phrase

• “if and only if”

Example

h: I watch footballk: the Eagles are playing

h iff kI watch football if and only if the Eagles

are playing

Different Forms of Conditional Statements

A conditional statement can be written in three different ways.

These three new conditional statements can be true or false.

EXAMPLE: pq If two angles are vertical, then they are congruent.

Converse: q pSWITCH (p and q but not if or then)

If two angles are congruent, then they are

vertical.

Inverse: ~p~qNEGATION (keep same order)

If two angles are not vertical, then they are

not congruent.

Contrapositive:~q~pSWITCH and NEGATE

If two angles are not congruent, then they are

not vertical.

Contrapositives are logically equivalent to the original

conditional statement.

If pq is true, then qp is true.

If pq is false, then qp is false.

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)

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