Analyze Conditional StatementsAnalyze Conditional Statements

Objectives:

1.To write a conditional statement in if-then form

2.To write the negation, converse, inverse, and contrapositive of a conditional statement and identify its truth value

3.To write a biconditional statement

Example 1Example 1

What are Clairzaps?

ConditionalsConditionals

ConditionalsConditionals are statements written in if-if-thenthen form.

A hexagon is a polygon with six sides.Subject Predicate

IfIf it is a hexagon, thenthen it is a polygon with six sides.

IfIf a polygon is a hexagon, thenthen it has six sides.

-OR- For clarity:

Hypothesis Conclusion

Example 2Example 2

Rewrite the conditional statement in if-then form.

All 90° angles are right angles.

Example 3Example 3

Rewrite the conditional statement in if-then form.

Two angles are supplementary if they are a linear pair.

ConverseConverse

The converseconverse of a conditional is formed by reversing the hypothesis (if) and conclusion (then).

Example 4Example 4

Write the following statement in if-then form, then write its converse. Is the converse always true?

All squares are rectangles.

Truth ValueTruth Value

A conditional statement can be truetrue or falsefalse.

• TrueTrue: To show that a conditional is true, you have to prove that the conclusion is true every time the hypothesis is satisfied.

• FalseFalse: To show a conditional is false, you just have to find one example in which the conclusion is not true when the hypothesis is satisfied.

Example 5Example 5

What is the opposite of the following statements?

1.The ball is red.

2.The cat is not black.

NegationNegation

The negationnegation of a statement is the opposite of the original statement.

Statement: Statement: The sick boy eats meat.

Negation: Negation: The sick boy does not eat meat.

Notice that only the verb of the sentence gets negated.

Symbolic NotationSymbolic Notation

Mathematicians are notoriously lazy, creating shorthand symbols for everything. Conditional statements are no different.

Symbol Concept

p Original Hypothesis

q Original Conclusion

→ “Implies”

~ “Not”

p → q “p implies q” “if p, then q”

~p “not p”

All Kinds of ConditionalsAll Kinds of Conditionals

So the symbols make conditionals easy and fun!

Statement Symbols

Conditional p → q

Converse q → p

Inverse ~p → ~q

Contrapositive ~q → ~p

All Kinds of StatementsAll Kinds of Statements

Here are some examples of writing the converse, inverse, and contrapositive of a conditional statement.

Example 6Example 6

Write the converse, inverse, and contrapositive of the conditional statement. Indicate the truth value of each statement.

If a polygon is regular, then it is equilateral.

Which of the statements that you wrote are equivalent?

Equivalent StatementsEquivalent Statements

When pairs of statements are both true or both false, they are called equivalent equivalent statementsstatements.

• A conditional and its contrapositive are equivalent.

• An inverse and the converse are equivalent.– So if a conditional is true, so its contrapositive.

Definitions in GeometryDefinitions in Geometry

In geometry, definitions can be written in if-then form. It is important that these definitions are reversiblereversible. In other words, the converse of a definition must also be true.

If a polygon is a hexagon, then it has exactly six sides.-AND-

If a polygon has exactly six sides, then it is a hexagon.

Perpendicular LinesPerpendicular Lines

If two lines intersect to form a right angle, then they are perpendicular perpendicular lineslines.

Example 7Example 7

Write the converse of the definition of perpendicular lines.

If two lines intersect to form a right angle, then they are perpendicular perpendicular lineslines.

BiconditionalBiconditional

A biconditionalbiconditional is a statement that combines a conditional and its true converse in “if and only if” form.

If a polygon is a hexagon, then it has exactly six sides.-AND-

If a polygon has exactly six sides, then it is a hexagon.

A polygon is a hexagon if and only if it has exactly six sides.

Example 8Example 8

Write the definition of perpendicular lines as a biconditional statement.

If two lines intersect to form a right angle, then they are perpendicular perpendicular lineslines.

Exercise 9Exercise 9

Rewrite the definition of right angle as a biconditional statement.