Logic
Logic questions don’t look like math questions at all;
they don’t contain numbers, formulas, or variables. Instead, logic
questions contain a verbal statement and a question that asks you
to interpret the validity of the statement or the effect of the
given statement on another statement. For example:
|
|
|
The
statement “If Jill misses the bus, she will be late” is true. Which
other statement must be true? |
| (A) |
If Jill does not miss the bus, she will not be late |
| (B) |
If Jill is not late, she missed the bus |
| (C) |
If a student misses the bus, he or she will be late |
| (D) |
Jill is late because she missed the bus |
| (E) |
If Jill is not late, she did not miss the bus |
|
Even though they are stated in words, logic questions
require mathematical methods for finding the right answers.
A logic statement is written in the form “If
p,
then
q,” where
p and
q are
events. “If
p, then
q” can also
be written as
,
and it states that if event
p occurs, then event
q will also
occur.
Every “If
p, then
q”
statement has an equivalent statement; this second statement is known
as the contrapositive, which is always true. The contrapositive
of “If
p, then
q” is “If not
q,
then not
p.” In symbols, the contrapositive of
is
(here, the symbol
~ means “not”). To formulate the contrapositive of any logic statement,
you must change the original statement in two ways.
-
Switch the order of the two parts of the statement.
For example, “If p, then q” becomes
“If q, then p.”
- Negate
each part of the statement. “If q, then p” becomes
“If not q, then not p.”
When faced with a logic problem on the Math IIC, remember
that if a given statement is true, then that statement’s contrapositive
is also true. Likewise, if a given statement is false, then that
statement’s contrapositive is also false.
Returning to the example problem, we are told that the
given statement is true, so we should look for the contrapositive
among the answer choices. E is the contrapositive of
the original statement, so we know that it is true. Here’s some
more practice:
|
|
|
What
is the contrapositive of “Every book on the shelf is old”? |
|
You need to first rewrite this statement so that it is
in the “If p, then q” form. So
the given statement becomes “If a book is on the shelf, then it
is old.” The contrapositive of the statement is now easy to see:
“If a book is not old, then it is not on the shelf.”
