Analyze%20Conditional%20Statements

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Analyze Conditional Statements

• Objectives
• To write a conditional statement in if-then form
• To write the negation, converse, inverse, and
  contrapositive of a conditional statement and
  identify its truth value
• To write a biconditional statement

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Example 1

• What are Clairzaps?

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Conditionals

• Conditionals are statements written in if-then
  form.

Subject
Predicate
A hexagon is a polygon with six sides.
If it is a hexagon, then it is a polygon with six
sides.
-OR- For clarity
If a polygon is a hexagon, then it has six sides.
Hypothesis
Conclusion
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Example 2

• Rewrite the conditional statement in if-then
  form.
• All 90 angles are right angles.

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Example 3

• Rewrite the conditional statement in if-then
  form.
• Two angles are supplementary if they are a linear
  pair.

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Converse

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

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Example 4

• Write the following statement in if-then form,
  then write its converse. Is the converse always
  true?
• All squares are rectangles.

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Truth Value

• A conditional statement can be true or false.
• True To show that a conditional is true, you
  have to prove that the conclusion is true every
  time the hypothesis is satisfied.
• False 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.

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Example 5

• What is the opposite of the following statements?
• The ball is red.
• The cat is not black.

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Negation

• The negation of a statement is the opposite of
  the original statement.
• Statement The sick boy eats meat.
• Negation The sick boy does not eat meat.
• Notice that only the verb of the sentence gets
  negated.

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Symbolic 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
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All 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
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All Kinds of Statements

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

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Example 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?

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Equivalent Statements

• When pairs of statements are both true or both
  false, they are called equivalent statements.
• A conditional and its contrapositive are
  equivalent.
• An inverse and the converse are equivalent.
• So if a conditional is true, so its
  contrapositive.

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Definitions in Geometry

• In geometry, definitions can be written in
  if-then form. It is important that these
  definitions are reversible. 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.
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Perpendicular Lines

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

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Example 7

• Write the converse of the definition of
  perpendicular lines.

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

• A biconditional 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.
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Example 8

• Write the definition of perpendicular lines as a
  biconditional statement.

If two lines intersect to form a right angle,
then they are perpendicular lines.
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Exercise 9

• Rewrite the definition of right angle as a
  biconditional statement.