Basic HTML Version
Q
uantitative
R
easoning &
P
roblem
S
olving
72
© 2014 Pacific Crest
2. Justify the following mathematical proof that “the square
of every even integer is even” by explaining why each
step is true.
Step
a. Let
x
be an even integer.
b. Then
x
= 2
y
for some integer
y
c.
x
2
= (2
y
)
2
d.
x
2
= 2(2
y
2
)
e. Therefore,
x
2
is even
W
hat Do You Already Know?
Tapping into your existing knowledge
1. How do you know that something that is “a given” is really true?
2. What is the difference between a theory and a proof?
3. What is the difference between an assumption and a premise?
4. What are four major differences between deductive reasoning and inductive reasoning?
5. How do you figure out if an argument or derivation is true?
M
athematical Language
Terms and notation
argument
— a set of logical steps
conclusion
— the end result of a proof
derivation
— a set of logical steps where you start with something and end with something else
fallacy
— an argument that uses unsound reasoning (there is a error in the reasoning)
logical fallacy
— an argument that violates the rules of logic
mathematical fallacy
— an argument or statement that violates the rules of mathematics
not equivalent
– two statements that are not logically equivalent
premise
— statements that start a proof
proof
— a set of logical steps to show that a statement is true
proposition
— a statement that is either true or false
Symbol
Meaning
Name
Definition
¬P, not P,
or
~P
NOT
negation
This logical operator negates a proposition.
Truth table
Examples of ¬A
P
¬P
True False
False True
➊
P: A quarter is equivalent to 5 nickels
¬
P: A quarter is not equivalent to 5 nickels
➋
P: 7 < 5
¬
P: 7 ≥ 5