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75
Symbol
Meaning
Name
Definition
A
⇔
B
or
A
≡
B
IF AND
ONLY IF
material
equivalence
A
⇔
B is true only if both A and B are true or both
A and B are false. The truth of A
⇔
B can vary from
one model or context to another; it is not universal.
Truth table
Examples of A
⇔
B
A
B A
⇔
B
True True True
True False False
False True False
False False True
➊
A: John passes his math test
B: John scores 65% on the test
John passes his math test
if and
only if
he scores 65% on the test.
➋
A: L > X
B: The symbolic values of L and X in Roman numerals
I
nformation
What you need to know
R
eadings
R
esources
M
ethodology
U
nderstanding a
D
erivation or
P
roof
Scenario:
Prove that 2
is irrational by the proposition that “ 2
is rational.”
Step
Explanation
Watch it Work!
1.
Clarify the
proposition
What is it that you want to prove,
derive, or disprove?
2
is rational
2.
List the premises
What are the givens that you start
with in order to derive or prove a
conclusion?
m
n
is a rational number
a
b
is reduced by a factor of 2 so
that either
a
or
b
is odd
3.
Identify the rules
used for each step
For each step, ensure that the step
is justified with a rule (logical or
mathematical)
The second column of the proof
includes the definitions of
even
,
and
function
, the multiplicative
identity, and property of exponents
4.
Validate that
conditions are met
For every rule, check to see that the
rule was used in appropriate context
There are no special conditions of
concern
5.
Verify appropriate
use of rules
Test to see if the rule was applied
correctly
Each step was executed correctly
6.
Does the
conclusion support
the proposition?
Did you end with the result that you
set out to achieve (i.e., is it proven or
is the derivation completed)?
The conclusion invalidates the
proposition, thus supporting the
conclusion desired.
2.2 Logical Reasoning