Basic HTML Version
Q
uantitative
R
easoning &
P
roblem
S
olving
74
© 2014 Pacific Crest
Symbol
Meaning
Name
Definition
A
→
B
IF...THEN
conditional
If A is true then B is also true. Note that this is not
a causal relationship; A does not cause B. The only
situation where the proposition is false is when A is
true but B is false (see Example 2).
Truth table
Examples of A → B
A
B A
→
B
True True True
True False False
False True True
False False True
➊
A: John misses more than 4 classes
B: John will fail the class
If
John misses more than 4 classes,
then
he will fail the class.
➋
A: 5
2
= 25
B: 8 is prime
A
B A
→
B
5
2
= 25
(True)
8 is prime
(False)
False
Expanded example of A → B
A: 10 or more people in your dining party
B: the restaurant charges a 15% gratuity
“
If
there are 10 or more people in your dining party
then
the restaurant charges a 15% gratuity.”
A does not cause B; A
→
B is simply a
proposition (policy) set by the restaurant.
A
B
A
→
B
1 people ≥ 10 15% gratuity True
2 people ≥ 10 no gratuity False
3 people < 10 15% gratuity True
4 people < 10 no gratuity True
Case 1:
If there are more than 10 in your dining party, then the restaurant follows its policy and
charges the gratuity (True → True: True)
Case 2:
If there are more than 10 in your dining party, then the restaurant fails to charge a gratuity. It
has not met its own policy of charging a gratuity on the condition of there being more than 10
in your party (the proposition is false). (True → False: False)
Case 3:
If there are fewer than 10 in your dining party then the restaurant charges the gratuity anyway.
The restaurant did not fail to meet its policy (it did
not
make the policy
false
, and therefore
the policy is true). The policy said nothing about
not
charging a gratuity if there were
fewer
than 10 in your party. That would have been a different proposition. (False → True: True)
Case 4:
If there are fewer than 10 in your dining party then the restaurant charges no gratuity. No
problem and a case of what we would normally call “fair play.” (False → False: True)
Symbol
Meaning
Name
Definition
A
+
B
EITHER...OR
exclusive
disjunction
A
+
B is true only when A or B, but not both are
true.
Truth table
Examples of A
+
B
A
B A
+
B
True True False
True False True
False True True
False False False
➊
A: The door is open
B: The door is closed
Either
the door is open
or
the door is closed.
➋
A: x < 0
B: x ≥ 0