GeometryIf-then Statements and Postulates

Section 2.2

If-then Statements and Postulates

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Babies are illogical.Nobody is despised who can manage a crocodile.Illogical persons are despised.

Lewis Carroll

Rewriting these in an If -then format helps to clarify the preceding statements.

If a person is not logical, then the person is despised.

If a person is a baby, then the person is not logical.

If a person is not despised, then that person can manage a crocodile.

What is the logical conclusion of these statements?

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If-then statements are called conditional statements or conditionals.

Hypothesis - is the If part (less the if), and the

Conclusion - is the then part (less the then) of the conditional.

If p, then q, where p and q are some statement, is represented symbolically with p q.

Symbolically, if p, then q (original), becomes p q

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Parallel lines don’t intersect.

How would you arrange this to make an if-then statement?

If lines are parallel, then they do not intersect.

Linear pairs are supplementary.

How would you arrange this to make an if-then statement?

If two angles are a linear pair, then they are supplementary angles.

Notice that all statements may not be an if-then statement format.

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When you are converting sentences to if-then statement, look for the key words, if and then. In our examples the if-part is always first, but some sentences may have the if-part at the end of the sentence. For example,

I will go to your house, if it rains tomorrow.

The hypothesis of this sentence is “it rains tomorrow” and the conclusion is “I will go to your house.” To make this a conditional

statement it would be rearranged as follows.

If it rains tomorrow, then I will go to your house.

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If a sentence has no if-then key words, then use the subject of the sentence as the hypothesis and the object of the sentence as the

conclusion. An example is;

Babies are illogical.

becomes;

If a person is a baby, then that person is illogical.

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The converse of a statement is when you exchange the hypothesis and the conclusion in a statement.

When p q, then the converse is q p.

If two lines are perpendicular, then they intersect.

and the converse is

If two lines intersect, then they are perpendicular.

Symbolically, p q (original), becomes q p

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Negation - The denial of a statement.

The angle is obtuse.

The denial of this statement is ;

Other examples of mathematical statements and then their denial

Symbolically, p (original), becomes p

The angle is not obtuse.

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The inverse of a statement is formed by negating the hypothesis and the conclusion.

Vertical angles are congruent.

and its inverse is,

If two angles are NOT vertical, then they are NOT congruent.

Symbolically, p q (original), becomes p q

If two angles are vertical, then they are congruent.

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The contrapositive of a statement is formed by negating the hypothesis and the conclusion of the converse.

Symbolically, p q (original), becomes q p

originalIf two lines are perpendicular, then they intersect.

converseIf two lines intersect, then they are perpendicular.

contrapositiveIf two lines do not intersect, then they are not perpendicular.

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It can be proven that the contrapositive is logically equivalent to the original statement.

A logically equivalent statement has the form of p q q p

orif x2 > 4, then x > 2

if (x > 2), then (x2 > 4)if x 2, then x2 4

if (3 = n + 1), then n = 2if (n=2), then (3 = n + 1)

if n 2, then 3 n + 1

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If you are 13 years old, then you are a teenager.

contrapositive

If you are NOT a teenager, then you are NOT 13 years old.

inverse **

If you are NOT 13 years old, then you are NOT a teenager.

converse **

If you are a teenager, then you are 13 years old.

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If an angle measures 90o, then it is a right angle.

contrapositive

If an angle does NOT measure 90o, then is NOT a right angle.

inverse

If an angle is NOT a right angle, then it does NOT measure 90o.

converse

If an angle is a right angle, then it measures 90o.

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SummaryWe converted written statements into conditionals (if-then

statements) in order to use logic to determine the validity (truth or falsehood) of those statements.

The form of the if-then is if p, then q, or p q, where the if-part of the conditional is the hypothesis and the then-part is the

conclusion.

We discussed the converse, the negation, the inverse, and the contrapositive of a conditional. We found that the inverse and the converse of a true conditional, is not always itself true. We also discovered that the conditional and its contrapositive are

logically equivalent.

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