G.1ab
Logic
Conditional Statements
Modified by Lisa Palen
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Conditional Statement
Defn. A conditional statement is a
statement that can be written as
an if-then statement. That is, as
“If _____________, then ______________.”
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Example:
If your feet smell
and your nose runs,
then you're built upside down.
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Conditional Statements have two parts:
The hypothesis is the part of a conditional
statement that follows “if” (when written in ifthen form.)
It is the given information, or the condition.
If a number is prime, then a number has
exactly two divisors.
Leave off “if” and
comma.
Hypothesis: a number is prime
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Conditional Statements have two parts:
The conclusion is the part of a conditional statement
that follows “then” (when written in if-then form.)
It is the result of the given information.
If a number is prime, then a number has
exactly two divisors.
Leave off “then”
and period
Conclusion: a number has exactly two divisors
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Rewriting Conditional Statements
Conditional statements can be put into an
“if-then” form to clarify which part is the
hypothesis and which is the conclusion.
Method: Turn the subject into a hypothesis.
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Example 1:
Vertical angles are congruent.
can be written as...
If two angles are vertical,
then they are congruent.
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Example 2:
Seals swim.
can be written as...
If an animal is a seal, then it swims.
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Example 3:
Babies are illogical.
can be written as...
If a person is a baby, then the person is
illogical.
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IF …THEN vs. IMPLIES
Another way of writing an if-then
statement is using the word implies.
Two angles are vertical
implies 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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Counterexample
Statement: If you live in Virginia, then
you live in Richmond, VA.
Is there a counterexample?
YES... Anyone who lives in Virginia, but
not Richmond, VA.
Therefore () the statement is false.
Symbolic Logic
Symbols can be
used to modify or
connect statements.
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Symbols for
Hypothesis and Conclusion
Lower case letters, such as p and q, are
frequently used to represent the hypothesis
and conclusion.
if p, then q
or
p implies q
Symbols for
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Hypothesis and Conclusion
Example
p: a number is prime
q: a number has exactly two divisors
if p, then q or p implies q
If a number is prime, then it
has exactly two divisors.
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
is used to represent the words
“if … then”
or
“implies”
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pq
means
if p, then q
or
p implies q
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Example
p: a number is prime
q: a number has exactly two
divisors
pq:
If a number is prime, then it
has exactly two divisors.
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~
is used to represent the word
“not”
•~ p is the negation of p.
•The negation of a statement is the
denial of the statement. Add or
remove the word “not.”
•To negate, write ~ p.
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Example
p: the angle is obtuse
~p: the angle is not obtuse
Be careful because ~p means
that the angle could be acute,
right, or straight.
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Example
p: James doesn’t like fish.
~p: James likes fish.
Notice: ~p 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
p: a number is even
q: a number is divisible by 3
pq: 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
p: a number is even
q: a number is divisible by 3
pq: A number is even
or it is divisible by 3.
2,3,4,6,8,9,10,12,14,15,...

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is used to represent the word
“therefore”
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Example
Therefore, the statement is
false.
 the statement is false
Different Forms of
Conditional Statements
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Forms of Conditional Statements
Converse: Statement formed from a conditional statement by
switching the hypothesis and conclusion (q  p)
pq If two angles are vertical, then they are congruent.
qp If two angles are congruent, then they are vertical.
Are these statements true or false?
Continued…..
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Forms of Conditional Statements
Inverse: Statement formed from a conditional statement
by negating both the hypothesis and conclusion.
(~p~q)
pq : If two angles are vertical, then they are congruent.
~p~q: If two angles are not vertical, then they are not
congruent.
Are these statements true or false?
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Forms of Conditional Statements
Contrapositive: Statement formed from a conditional statement
by switching and negating both the hypothesis and conclusion.
(~q~p)
pq : If two angles are vertical, then they are congruent.
~q~p: If they are not congruent, then two angles are not
vertical
Are these statements true or false?
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Contrapositives are logically equivalent to the original
conditional statement.
If
pq is true,
then qp is true.
If
pq is false,
then qp is false.
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Biconditional

When a conditional statement and its
•
•
•
•
converse are both true, the two statements
may be combined.
A statement combining a conditional
statement and its converse is a biconditional.
Use the phrase if and only if which is
abbreviated iff
Use the symbol

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Definitions are always biconditional

Statement: pq

If an angle is right then it measures 90.

Converse: qp

If an angle measures 90, then it is right.

Biconditional: pq

An angle is right iff it measures 90.
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Biconditional 
A
biconditional is in the form:
Hypothesis if and only if Conclusion.
or
Hypothesis iff Conclusion
or
Hypothesis

Conclusion
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Biconditionals in symbols
Since p  q
means pq AND qp,
pq
Is equivalent to
(pq)  (qp)