3.1 Boolean Operators

There are times in programs where you will need to make more complex decisions. You may need to make a choice based on the value of more than one variable. There are three boolean operators that you can use to make more complex boolean expressions. The three operators are and, or, and not.

Suppose you had the following declaration:

var p, q : boolean

p and q will only be true if both p and q are true. If either one is false then the result will be false.
p or q will be true if either p or q or both are true. It will only be false if both are false.
not p will be the opposite of p. If p is true the result will be false. If p is false the result will be true.

These boolean operators can be summarized in the following truth table:

p q p and q p or q not p

true true true true false

true false false true false

false true false true true

false false false false true

Each of these operators has a different precedence in the order of operations. The operator with the highest precedence of the three is not. The next highest is and. The lowest is or. Here is a table that outlines the order of operations for all the operators we've seen so far. Operators on the same line have the same precedence.

** * / div mod + - < <= = not= > >= not and or

Here are some examples using boolean operators:

  not false and true
= true and true
= true

  5 < -10 or 'M' >= 'M'
= false or true
= true

  not (5 + 10 = 15 and 7 not= 8)
= not (15 = 15 and true)
= not (true and true)
= not true
= false

  false or true and false
= false or false
= false

Take note of the last example. Remember that ands are executed before ors. If you do the or first you will get the wrong answer. Here is a small program using two of the new operators:

var age : int put "Please enter your age" get age % negative ages or ages that are too old are not possible if age < 0 or age > 120 then put "Do you really expect me to believe you are that age" end if % ages 13, 14, 15,16, 17, 18, 19 are teen ages if age >= 13 and age <= 19 then put "You are a teenager" end if

Exercise 3.1

Evaluate the expressions that are legal. If an expression is not legal, state the reason for its being illegal.

true and not false

false or not true

not (true or not false)

not (4 not= 2 + 2)

not 5 < 7

not (5 <= 8) or not false

Variables p, q, r, s are declared as:
var p, s : boolean := true var q, r : boolean := false
State the value of each expression.

not p

p or q

p and r

not (q and s)

not q and s

s and not q

p or not s

not p and not q

s or (not q and r)

p = (q or r)

Boolean expressions can often be simplified using a pair of rules known as de Morgan's laws. These laws state that:
M1: not (p and q) is equivalent to (not p) or (not q)
M2: not (p or q) is equivalent to (not p) and (not q)
Use de Morgan's laws and the fact that not(not p) is equivalent to p to simplify each expression.

not (p and not q)

not (not p and not q)

not (not p or not q)

not ((x not= 0) and not (y = 0))

The exclusive or operator, xor, has the following truth table.

p q p xor q

true true false

true false true

false true true

false false false

Using only and, or, not write an expression that is true if and only if p xor q is true.

p	q	p and q	p or q	not p
true	true	true	true	false
true	false	false	true	false
false	true	false	true	true
false	false	false	false	true