12.1 Logic
12.2 Sequences
12.3 Limits
12.4 Imaginary and Complex Numbers
12.5 Key Terms
12.6 Review Questions
12.7 Explanations
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.
  1. Switch the order of the two parts of the statement. For example, “If p, then q” becomes “If q, then p.”
  2. 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.”
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