Download Logic - Humble ISD

Get Ready To Be Logical!
• 1. Books and notebooks out.
• 2. Supplies ready to go.
• 3. Answer the following:
The sum of 2 positive integers is ___________
True or False
3
n
For all integers n, is positive.
2 complementary angles cannot be  .
Logic
Inductive Reasoning is a process of reasoning that a
rule or statement is true because of a specific case
that is usually drawn from a pattern or observation.
Examples
1. 1, 2, 4, 7, 11, _____
2.
Jan, March, May, ______
Conjecture
• Deductive Reasoning is a process of using logic
to draw conclusions using definitions, facts, or
properties. This may include postulates and
theorems.
Conditional
If p, then q.
p is hypothesis q is conclusion
p→q
If I work hard, then I will succeed.
Converse
If q, then p. Flip
q→p
If I will succeed, then I worked hard.
Conditional
If p, then q.
If I work hard, then I will succeed.
p→q
Inverse
If not p, then not q
negate ~p→~q
If I don’t work hard, then I will not succeed.
Contrapositive
If not q, then not p flip & negate ~q→~p
If I don’t succeed, then I didn’t work hard.
True or False
That is the Question???
To prove a statement false, one must provide a
counterexample.
A counterexample is a drawing, statement or
number. The counterexample must satisfy the
hypothesis but fail to satisfy the conclusion.
1. Look for a pattern
2. Make a conjecture
3. Prove or find a counterexample
Prove or find a counterexample
3
For all integers n, n is positive.
2 complementary angles cannot be  .
Truth value-- is true in all situations except when
hypothesis is true and the conclusion is false.
p = you make an A
q = I will buy you a car
p q
p→q
T T You made an A, then I bought the car.
T
T F You made an A, but I did not buy the car.
F
F T You did not make an A, but I bought the car anyway. T
F F You did not make an A, then I did not buy the car. T
Note --satisfying the hypothesis and failing to satisfy the conclusion is
what a counterexample does.
Note—counterexamples are used to prove statements false!
Write the converse, inverse, and contrapositive
of the following.
State if true or false.
If false give counterexample.
If m<A = 30, then <A is acute.
If m<A = 30, then <A is acute.
p
→
q
Converse
q
→
p
If <A is acute, then m<A = 30.
Inverse
~p → ~q
If m<A ≠30, then <A is not acute.
Contrapositive
~q → ~p
If <A is not acute, then m<A ≠ 30.
Write the converse, inverse, and contrapositive
of the following.
State if true or false.
If false give counterexample.
If 2 angles are vertical, then they are 
If 2 angles are vertical, then they are
p
→

q
Converse
q
→
p
If 2 angles are , then they are vertical.
Inverse
~p → ~q
If 2 angles are not vertical, then they are not 
Contrapositive
~q → ~p
If 2 angles are not  , then they are not vertical.
Biconditionals
• A biconditional is a blending of the conditional
statement and the converse.
• A biconditional may only be written if BOTH the
conditional AND the converse are true.
• A biconditional is in the form of p if and only if q.
This may be abbreviated p iff q or p
q.
• The statement --If an angle is a right angle, then it measures 90 °
This statement is true.
• The converse—If an angle measures 90°, then it is
a right angle.
This converse is true.
• Therefore, we can write a biconditional.
An angle measures 90° if and only if it is a right
angle.