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Conditional Statements
Lesson 2-1
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Conditional Statements have two
parts:
•Hypothesis (denoted by p) and
•Conclusion (denoted by q)
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Hypothesis (p)
•Phrase following “if”
•the given information
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Conclusion (q)
•Phrase following “then”
•the result of the given information
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Conditional statements can be
put into an “if-then” form to clarify which part is the hypothesis and which is the
conclusion.
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Example:Vertical angles are congruent.
If two angles are vertical, then they are congruent.
can be written as...
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If two angles are vertical,
then they are congruent.
p implies q
Hypothesis (p):two angles are vertical
Conclusion (q):they are congruent
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Conditional Statements
can be true or false:•A conditional statement is false only when the hypothesis is true, but the conclusion is false.
•A counterexample is an example used to show that a statement is not always true and therefore false.
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Giving a Counterexample
Therefore () the statement is false.
Statement: If you live in Virginia, then you live in Richmond.
True or False? Give a counterexample:
Henry lives in Virginia, BUT he lives in Ashland.
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Symbolic Logic
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Symbols can be used to modify or connect
statements.
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is used to represent the word
• “therefore”
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Example
H : I watch football
H
Therefore, I watch football
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is used to represent
• implies • used in if … then
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Example
p: a number is prime
q: a number has exactly two divisors
pq: If a number is prime, then it has exactly two divisors.
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~ is used to represent the word
• “not” or
• “negate”
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Example
w: the angle is obtuse
~w: the angle is not obtuse
Be careful because ~w means that the angle could be acute, right, or straight
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Example
r: I am not happy
~r: I am happy
Notice: ~r took the “not” out… it would have been a double negative (not not)
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is used to represent the word
• “and”
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Example
a: a number is even
b: a number is divisible by 3
ab: A number is even and it is divisible by 3.
6,12,18,24,30,36,42...
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is used to represent the word
• “or”
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Example
a: a number is even
b: a number is divisible by 3
ab: A number is even or it is divisible by 3.
2,3,4,6,8,9,10,12,14,15,...
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iff is used to represent the phrase
• “if and only if”
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Example
h: I watch footballk: the Eagles are playing
h iff kI watch football if and only if the Eagles
are playing
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Different Forms of Conditional Statements
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A conditional statement can be written in three different ways.
These three new conditional statements can be true or false.
EXAMPLE: pq If two angles are vertical, then they are congruent.
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Converse: q pSWITCH (p and q but not if or then)
If two angles are congruent, then they are
vertical.
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Inverse: ~p~qNEGATION (keep same order)
If two angles are not vertical, then they are
not congruent.
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Contrapositive:~q~pSWITCH and NEGATE
If two angles are not congruent, then they are
not vertical.
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Contrapositives are logically equivalent to the original
conditional statement.
If pq is true, then qp is true.
If pq is false, then qp is false.
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Biconditional• If a conditional statement and its
converse are both true, then the two statements may be combined.
• using the phrase if and only if (iff)
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Definitions are always biconditional
• Statement: If an angle is right then it has a measure of 90.
• Converse: If an angle measures 90, then it is a right angle.
• Biconditional: An angle is right iff it measures 90