Logic and Math

Chapter 4

Everything is a Number

Remember, to a computer, everything is a number, even characters.

Truth Tables

A convenient and helpful way to organize the possible values of any logical (Boolean) statement is in a truth table. A truth table is a table whose column headings are expressions, and whose rows are possible scenarios given the expressions. For example, in the NOT table below, if p is true (T), then “not p” (!p) must be false (e.g., if p means it “is raining” then !p means it is “not raining”).

Truth table for the NOT function.

p	!p
T	F
F	T

Truth table for the AND function.

p	q	p && q
T	T	T
T	F	F
F	T	F
F	F	F

Truth table for the OR function.

p	q	p || q
T	T	T
T	F	T
F	T	T
F	F	F

Examples

Truth table for ! (p || q) with intermediate step.

p	q	p || q	! (p || q)
T	T	T	F
T	F	T	F
F	T	T	F
F	F	F	T

Truth table for !p && !q with intermediate steps.

p	q	!p	!q	!p && !q
T	T	F	F	F
T	F	F	T	F
F	T	T	F	F
F	F	T	T	T

Note that !(p || q) has the same truth-table results as !p && !q. That means that the two statements are logically equivalent.

Operator Precedence

• Relational and logical operators are evaluated from left to right
  • The associativity is left to right
• Parentheses can override precedence

Operators	Precedence
!, +, - (unary operators)	first
*, /, %	second
+, -	third
<, <=, >=, >	fourth
==, !=	fifth
&&	sixth
||	seventh
=	last

Try to create more advanced combinations for truth tables like:

q || !q && p != q

What does this evaluate to?

1 + 7 < 2 || !5 + -1 && true == 0

Remember false is 0 and true is a nonzero value.

Short-Circuit Evaluation

Short-circuit evaluation is the evaluation of a logical expression that stops as soon as the value of the expression is known.

• Example:

  (age >= 21) || ( x == 5) //Line 1  
  (grade == 'A') && (x >= 7) //Line 2

• Example to prevent dividing by zero.

  int demon = 1, num = 4;  
  if(denom != 0 && num / denom > 0)  
      cout << num << " has the same sign as " << denom << endl;

Trick, trick question: If the value of num is 0, what is it after: if(num && num = 2) …?

Answer: num would still equal 0. But it would equal 2 if you changed the && to ||.

Self-Check Questions

1. Be able to fill in the truth tables.
2. Use truth tables to determine if two expressions are logically equivalent.
3. Does the following expression evaluate to true or false?
   • 1 + 2 < 10 || (4 * 2 != 9 && -4 / 2 <= -4)