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Section 2-3: Biconditionals & Definitions

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Section 2-3: Biconditionals& Definitions

- Objectives:
- Write Biconditionals and recognize good definitions

Recall

Form of a Conditional Statement

Biconditional

p q

- Write a bi-conditional only ifBOTH the conditional and the converse are TRUE.
- Connect the conditional & its converse with the word “and”
- Write by joining the two parts of each conditional with the phrase “if and only if” of “iff” for shorthand.
- Symbolically: p q

Bi-conditional Statements

Conditional Statement:

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

Converse:

If two angles are congruent, then they have the same measure.

Both statements are true, so….

…you can write a Biconditional statement:

Two angles have the same measure if and only if the angles are congruent.

Try It!

Write a Bi-conditional Statement

Consider the following true conditional statement. Write its converse. If the converse is also true, combine the statements as a biconditional.

Conditional:

If x = 5, then x + 15 = 20.

Converse:

If x + 15 = 20, then x = 5.

Since both the conditional and its converse are true, you can combine them in a true biconditional using the phrase if and only if.

Biconditional:

x = 5 if and only if x + 15 = 20.

Separate a Biconditional

- Write a biconditional as two conditionals that are converses of each other.

Consider the biconditional statement:

A number is divisible by 3 if and only if the sum of its digits is divisible by 3.

Statement 1:

If a number is divisible by 3, then the sum of its digits is divisible by 3.

Statement 2:

If the sum of a numbers digits is divisible by 3, then the number is divisible by 3.

Try It!

Separate a Biconditional

Write the two statements that form this biconditional.

Biconditional:

Lines are skew if and only if they are noncoplanar.

Conditional:

If lines are skew, then they are noncoplanar.

Converse:

If lines are noncoplanar, then they are skew.

Writing Definitions as Biconditionals

- Good Definitions:
- Help identify or classify an object
- Uses clearly understood terms
- Is precise avoiding words such as sort of and some
- Is reversible, meaning you can write a good definition as a biconditional (both conditional and converse are true)

Show definition of perpendicular lines is reversible

Definition:

Perpendicular lines are two lines that intersect to form right angles

Conditional:

If two lines are perpendicular, then they intersect to form right angles.

Converse

If two lines intersect to form right angles, then they are perpendicular.

Since both are true converses of each other, the definition can be written as a true biconditional:

“Two lines are perpendicular iff they intersect to form right angles.”

Writing Definitions as Biconditionals

Show that the definition of triangle is reversible. Then write it as a true biconditional.

Steps

1. Write the conditional

2. Write the converse

3. Determine if both statements are true

4. If true, combine to form a biconditional.

Definition: A triangle is a polygon with exactly three sides.

True

Conditional:

If a polygon is a triangle, then it has exactly three sides.

True

Converse:

If a polygon has exactly three sides, then it is a triangle.

Biconditional:

A polygon is a triangle if and only if it has exactly three sides.

Writing Definitions as Biconditionals

Is the following statement a good definition? Explain.

An apple is a fruit that contains seeds.

Conditional: If a fruit is an apple then it contains seeds.

Converse: If a fruit contains seed then it is an apple.

There are many other fruits containing seeds that are not apples, such as lemons and peaches. These are counterexamples, so the reverse of the statement is false.

The original statement is not a good definition because the statement is not reversible.

Summary