Conditional statements relate two parts, a hypothesis and conclusion, using "if-then" logic. The hypothesis follows "if" and is the given condition, while the conclusion follows "then" and is the result. Symbols like implies (®), not (~), and (Ù,Ú) can also represent conditional statements and connect multiple statements. A conditional statement is only false if the hypothesis is true but conclusion is false. Contrapositives and converses of conditional statements are also considered in determining logical equivalencies.

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Definition: A conditional statement is a statement that
can be written in if-then form.
“If _____________, then ______________.”
Example: If your feet smell and your nose runs, then
you're built upside down.
Continued……

3. Conditional Statements have two parts:
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The hypothesis is the part of a conditional statement that follows
“if” (when written in if-then form.)
The hypothesis is the given information, or the condition.
The conclusion is the part of an if-then statement that follows
“then” (when written in if-then form.)
The conclusion is the result of the given information.

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Writing Conditional Statements
Hint: Turn the subject into the hypothesis.
Example 1: Vertical angles are congruent. can be written as...
Conditional
Statement:
Example 2: Seals swim. can be written as...
Conditional
Statement: If an animal is a seal, then it swims.
If two angles are vertical, then they are congruent.

5. Two angles are vertical implies they are congruent.
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Another way of writing an if-then statement is using
the word implies.
If two angles are vertical, then they are congruent.

6.  A conditional statement is false only when the hypothesis is
true, but the conclusion is false.
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A counterexample is an example used to show that a
statement is not always true and therefore false.
If you live in Virginia, th Statement: en you live in Richmond.
Is there a counterexample?
Yes !!!
Counterexample: I live in Virginia, BUT I live in Glen Allen.
Therefore () the statement is false.

7.  Symbols can be used to modify or connect statements.
 Symbols for Hypothesis and Conclusion:
Hypothesis is represented by “p”.
Conclusion is represented by “q”.
if p, then q
or
p implies q
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Continued…..

8. if p, then q
or
p implies q
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p ® q is used to represent
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.
Continued…..

9. ~p:
Note:
Example 2: p: I am not happy
~p: I am happy
~p took the “not” out- it would have been a double negative (not not)
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Symbolic Logic - continued
~ “not”
Example 1: p: the angle is obtuse
The angle is not obtuse
~p means that the angle could be acute, right, or straight.

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Symbolic Logic - continued
Ù “and”
Example: p: a number is even
q: a number is divisible by 3
A number is even and it is divisible by 3.
i.e. 6,12,18,24,30,36,42...
pÙq:

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Symbolic Logic- continued
Ú “or”
Example: p: a number is even
q: a number is divisible by 3
pÚq: A number is even or it is divisible by 3.
i.e. 2,3,4,6,8,9,10,12,14,15,...

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“therefore”
Symbolic Logic - continued
Example: Therefore, the statement is false.
the statement is false

13. Converse: Switch 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.
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Continued…..

14. Inverse: State the opposite of 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.
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15. Contrapositive: Switch the hypothesis and conclusion and
state their opposites. (~q®~p)
p®q : If two angles are vertical, then they are congruent.
~q®~p: If two angles are not congruent, then they are not
vertical.
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16.  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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17.  When a conditional statement and its converse are both true,
the two statements may be combined.
 Use the phrase if and only if (sometimes abbreviated: iff)
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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 if and only if it measures 90°.

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