Basic HTML Version
Q
uantitative
R
easoning &
P
roblem
S
olving
76
© 2014 Pacific Crest
O
ops
! A
voiding
C
ommon
E
rrors
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Assuming that if you have A or B, and you have A, then you don’t have B.
Example
: The man is rich or handsome. He is handsome so he must not be rich.
Why?
The statement “A or B” does not mean that an item
cannot be both; it is a simple disjunction (not exclusive).
Therefore if A is true then B can also be true.
The general
truth table for the proposition it at right; an expanded
and specific truth table is shown below.
A
(OR) B
A ˅ B
the man is rich
the man is handsome
True
the man is rich the man is NOT handsome True
the man is NOT rich the man is handsome
True
the man is NOT rich the man is NOT handsome False
A
B A ˅ B
True True True
True False True
False True True
False False False
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If A, then B; B, therefore A
Example
: If it is raining, I will have an umbrella. Just because I have an umbrella does not mean
it is raining.
Why?
If A implies B and you have B it does not mean that you
also have A.
The general truth table for the proposition
is at right; an expanded and specific truth table is shown
below.
(IF) A
(THEN) B
A
→
B
it is raining
I will have an umbrella
True
it is raining I will NOT have an umbrella False
it is NOT raining I will have an umbrella
True
it is NOT raining I will NOT have an umbrella True
A
B A
→
B
True True True
True False False
False True True
False False True
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Misapplying mathematical rules (mathematical fallacy)
Example
: Theorem: $1 = 1¢
Proof:
Step 1 $1 = 100¢
Step 2 $1 = (10¢)
2
Step 3 $1 = (0.1$)
2
Step 4 $1 = 0.01$
Step 5 $1 = 1¢
Why?
The results of a mathematical proof are only valid when the rules of mathematics
have been correctly, accurately, and rigorously followed. In this instance, Step 2 is
incorrect. 100¢ = (10)
2
¢
not
(10¢)
2
. Because the proof violated the Laws of Exponents
in Step 2, this proof is invalid.