Conditional Statements

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Conditional Statement-A logical statement with two

parts, an "If" portion called the hypotheses and a "then"portion

called a conclusion.

Ex. If it is sunny outside, then it is warm.

NOTATION: p q

This is read "If p then q"

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Circle the hypothesis and underline the conclusion of

the following Conditional Statements.

1. If it is an odd number, then it is divisible by three.

2. If x=3, then 2x+6=12.

Make the following conjecture into a conditional

statement.

1. When x=6, x

2=36.

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Truth Values

Just like a conjecture, a conditional statement can be always true or false.

This is known as the truth value.

A statement is False if there exists at least one counter example- or at least

one time when the statement can be proven false.

Ex: If it is warm outside, then it is June.

Counter Example: It could be May.

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Converse

The converse of a conditional statement is the switching of the

hypothesis and the conclusion.

Ex: Conditional Statement: If I wear converses, then I wear High Tops

Converse: If I wear High Tops, then I wear converse.

p

q

p

q

Notation: q p

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Find the converse of the conditional statement, then decide the

truth value. If False, provide a counter example.

1. If an angle is obtuse, then its measure is over 90 degrees.

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Inverse

Negate both the hypothesis and the conclusion

*Negate means to "make negative" *The notation for "not" is a ~

Ex: Conditional Statement: If an angle is 90 degrees, then it is a right

angle.

Inverse: If an angle is not 90 degrees, then it is not a right angle.

p

q

~p

~q

Read "If not p, then not q"

Notation: ~p ~q

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Find the inverse of the conditional statement, then decide the

truth value. If False, provide a counter example.

1. If x=3 , then x+2=5.

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Contrapositive

Negate both the hypothesis and the conclusion of the converse.

or just Negate then switch the conditional.

Ex: Conditional Statement: If an angle is 90 degrees, then it is a right angle.

Contrapositive: If an angle is not a right angle, then it is not 90 degrees.

p

q

~q

~p

Read "If not q, then not p"

Notation: ~q ~p

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Find the contrapositive of the conditional statement, then decide

the truth value. If False, provide a counter example.

1. If I do not have my ID on, then I get hours.

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Biconditional

A statement that combines a true conditional statement and a true converse.

We do this by replacing the "if" and "then" with the phrase "if and only if".

Ex:
Conditional: If two angles have the same measure, then the angles are
congruent.

Converse: If two angles are congruent, then the two angles have the same

measure.

Biconditional: Two angles have the same measure if and only if the angles are

congruent.

Notation: p q

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Determine the truth values of both statements

Conditional: If two angles have the same measure, then the

angles are congruent.

Converse: If two angles are congruent, then the angles are

the same measure.

Are these both true? How can we combine the two

statements to make them both true all of the time?

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Summary:

Conditional statement

If p then q

p q

Converse

If q then p

q p

Inverse

If not p then not q

~p ~q

Contrapositive

If not q then not p

~q ~p

Biconditional

p iff (if and only if) q p q

Statement Type How it is read

Notation

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Given the following conjecture, create the conditional statement, converse,

inverse, and contrapositive

x is an even number, x can be divided by 2.

Conditional Statement:

Converse:

Inverse:

Contrapositive:

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