2-3 Conditional StatementsMs. Hinrichsen

“If you’re not making mistakes, then you’re not doing anything. I’m positive that a doer makes

mistakes.”- John Wooden

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• A conditional statement is a statement that can be writtenin if-then form.

• Hypothesis: the phrase followingthe word if

• Conclusion: the phrase following theword then

• Notation: p q• (read “p implies q”)

•Conditional Statements

Hypothesis: it rains

Conclusion: the grass is wet

If it rains, then the grass is wet.

• Example 1 Identify the hypothesis and conclusion for each conditional statement.

• If it’s Sunday, then football will be on TV.• Hypothesis: it’s Sunday• Conclusion: football will be on TV

• A number is even if it is divisible by two.• Hypothesis: it is divisible by two• Conclusion: a number is even

•Conditional Statements

A student will lose credit for a class after being tardy 18 times.

• Hypothesis: being tardy 18 times

• Conclusion: a student will lose credit for a class

•Rewriting Conditional Statements

• Not all conditional statements are written using if-then form. To rewrite the statement in if-then form, you need to identify the hypothesis and conclusion.

• WARNING: The rewrite of the statement in if-then form is not always a word-for-word translation of the original statement.

• “A student will lose credit for a class after being tardy 18 times.”

• can be rewritten as

• “If a student is tardy 18 times, then the student will lose credit for the class.”

•Rewriting Conditional Statements

a) A golden retriever is a dog.

• If an animal is a golden retriever, then it is a dog.

• Hypothesis: an animal is a golden retriever

• Conclusion: it is a dog

b) A hexagon has six sides.

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

• Hypothesis: a polygon is a hexagon

• Conclusion: it has six sides

•Rewriting Conditional StatementsExample 2 Rewrite each statement in if-then form. Identify the hypothesis and conclusion.

• Converse• Switches the hypothesis and conclusion• Notation: q p

• Inverse• Negates the hypothesis and conclusion• Notation: ~p ~q

• Contrapositive• Negates and switches the hypothesis and conclusion• Notation: ~q ~p

•Related Conditional Statements

•Related Conditional StatementsIf an angle measures 16°, then the angle is acute.

Converse: If the angle is acute, then an angle measures 16°.

(switched p and q)

Inverse: If an angle does not measure 16°, then the angle is not acute.

(negated p and q – made them opposite)

Contrapositive: If the angle is not acute, then an angle does not measure 16°.

(negated and switched p and q)

•Related Conditional StatementsConditional: If an angle measures 16°, then the angle is acute.

Converse: If the angle is acute, then an angle measures 16°.

Inverse: If an angle does not measure 16°, then the angle is not acute.

Contrapositive: If the angle is not acute, then an angle does not measure 16°.

Think about which statements are true or false. Notice anything????

•Related Conditional StatementsA conditional and its contrapositive are logically equivalent.

The converse and the inverse are logically equivalent.

What does it mean to be logically equivalent?

For statements to be logically equivalent means that they have the same truth value. In other words –

both statements are true or both are false.

• Example 3 For the given conditional statement, write the converse, inverse, and the contrapositive.

“If two lines are skew, then the lines are not coplanar.”

Converse: If the lines are not coplanar, then two lines are skew.

Inverse: If two lines are not skew, then the lines are coplanar.

Contrapositive: If the lines are coplanar,then two lines are not skew.

•Related Conditional Statements

•Related Conditional Statements• Example 4 Write the statement in if-then form and

then write the • converse, inverse, and contrapositive.• “An integer that ends in zero is divisible by five.”

Conditional: If an integer ends in zero, then it is divisible by five.Converse: If it is divisible by five, then an integer ends in zero.Inverse: If an integer doesn’t end in zero, then it isn’t divisible by five.Contrapositive: If it isn’t divisible by five, then an integer doesn't end in zero.

Think about the statements that are supposed to be logically equivalent. Are they? Reading skills play a

tremendous role in logic discussions!

• Example 5 “If it is a rectangle, then the polygon is a square” is the converse of a conditional statement. Find the inverse of the original conditional and explain which statements are true or false.

Given: Converse q p Need: Inverse ~p ~q

Inverse: If the polygon is not a square, then it is not a rectangle.

Both statements are false. This makes sense since the converse and the inverse are logically equivalent.

•Related Conditional Statements

• A biconditional statement is a compound statement of a conditional statement and its converse joined using the word “and”.

•Biconditional Statements

Example:

An angle is a right angle if and only if its

measure is 90°.

p q if and only if q p

p q iff q p

(iff = if and only if)

(p q) ∧ (q p)

• A biconditional statement can be rewritten as a conditional and its converse.

• An angle is a right angle if and only if its measure is 90°.

• Conditional: If its measure is 90°, then an angle is a right angle.

• Converse: If an angle is a right angle, then its measure is 90°.

• A biconditional statement is true only when both the conditional and the converse are true.

•Biconditional Statements

• Example 6 Write the biconditional statement as a conditional and its converse. Determine whether the statement is is true or false.

• x2 is positive if and only if x is real number.

• Conditional: If x is a real number, then x2 is positive.

• Converse: If x2 is positive, then x is a real number.

• The biconditional statement is false.

•Biconditional Statements