analyze conditional statements objectives: 1.to write a conditional statement in if-then form 2.to...
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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
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Example 1Example 1
What are Clairzaps?
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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
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Example 2Example 2
Rewrite the conditional statement in if-then form.
All 90° angles are right angles.
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Example 3Example 3
Rewrite the conditional statement in if-then form.
Two angles are supplementary if they are a linear pair.
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ConverseConverse
The converseconverse of a conditional is formed by reversing the hypothesis (if) and conclusion (then).
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Example 4Example 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 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.
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Example 5Example 5
What is the opposite of the following statements?
1.The ball is red.
2.The cat is not black.
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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.
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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”
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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
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All Kinds of StatementsAll Kinds of Statements
Here are some examples of writing the converse, inverse, and contrapositive of a conditional statement.
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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?
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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.
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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.
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Perpendicular LinesPerpendicular Lines
If two lines intersect to form a right angle, then they are perpendicular perpendicular lineslines.
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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.
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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.
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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.
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Exercise 9Exercise 9
Rewrite the definition of right angle as a biconditional statement.