Mathematics for Computer Science

3.2. Propositional Logic in Computer Programs 41

easier to read and understand, and can also make it faster since fewer operations
are needed. In hardware, simplifying expressions can decrease the number of logic
gates on a chip. That’s because digital circuits can be described by logical formu-
las (see Problems 3.5 and 3.6), and minimizing the logical formulas corresponds
to reducing the number of gates in the circuit. The payoff of gate minimization is
potentially enormous: a chip with fewer gates is smaller, consumes less power, has
a lower defect rate, and is cheaper to manufacture.

3.2.2 Cryptic Notation

Java uses symbols like “&&” and “jj” in place ofANDandOR. Circuit designers
use “
” and “C,” and actually refer toANDas a product andORas a sum. Mathe-
maticians use still other symbols given in the table below.

English Symbolic Notation

NOT.P/ :P (alternatively,P)

PANDQ P^Q

PORQ P_Q

PIMPLIESQ P!Q

ifPthenQ P!Q

PIFFQ P !Q

PXORQ P ̊Q

For example, using this notation, “IfPAND NOT.Q/, thenR” would be written:

.P^Q/!R:

The mathematical notation is concise but cryptic. Words such as “AND” and
“OR” are easier to remember and won’t get confused with operations on numbers.
We will often usePas an abbreviation forNOT.P/, but aside from that, we mostly
stick to the words—except when formulas would otherwise run off the page.

Exam Problems

Problem 3.1.
Show that there are exactly two truth assignments for the variables P,Q,R,S that
satisfy the following formula:

.PORQ/AND.QORR/AND.RORS/AND.SORP/
Hint:A truth table will do the job, but it will have a bunch of rows. A proof by
cases can be quicker; if you do use cases, be sure each one is clearly specified.