7.4 Boolean Expressions, Truth Tables, and Boolean Algebra

Q: Which expression is equivalent to `not ( ( temp ≥ 32 ) and ( weather ≠ "snow" ) )`?

`( temp < 32 ) or ( weather == "snow" )`. By De Morgan's law, not ( P and Q ) = not P or not Q. The negation of temp ≥ 32 is temp < 32, and the negation of weather ≠ "snow" is weather == "snow". Joining them with or gives the first choice. Keeping and, or negating a comparison incorrectly, produces the other choices.

Q: Which single condition makes the following code print "both" in exactly the same situations? ``` if ( x > 0 ) if ( y > 0 ) print "both" end if end if ```

`if ( ( x > 0 ) and ( y > 0 ) )`. The inner if is reached only when x > 0, and it prints only when y > 0 as well, so the combined condition is ( x > 0 ) and ( y > 0 ). Using or would print when only one is positive, and changing > to ≥ would print when x or y is 0.

Q: Which condition is true exactly when `x` is **outside** the range 10 to 20, inclusive?

`( x 20 )`. "Inside" is ( x ≥ 10 ) and ( x ≤ 20 ). Its negation, by De Morgan, is ( x 20 ). Joining the two comparisons with and can never be true. Using ≤ and ≥ wrongly treats 10 and 20 as outside, and the third choice describes the inside of the range.

Q: An array `list` has valid indexes 0 through n − 1. Which condition safely checks whether the element at index `i` equals `target` when `i` might equal n, assuming short-circuit evaluation?

`( i < n ) and ( list[i] == target )`. With and evaluated left to right and short-circuited, placing i < n first means list[i] is evaluated only when i is a valid index. When i = n, the left side is false and evaluation stops. The first two choices read list[n] before any check. The last choice evaluates list[i] precisely when i < n is false, which is the invalid case.

Key Takeaways

and is true only when both operands are true; or is true when at least one is true; not reverses a value; a truth table for n variables has 2ⁿ rows.
De Morgan's laws: not ( A and B ) equals ( not A ) or ( not B ), and not ( A or B ) equals ( not A ) and ( not B ).
Negating a comparison flips it: not ( x > 10 ) is x ≤ 10, and not ( x == y ) is x ≠ y.
Nested if statements that both must pass are equivalent to one condition joined with and; alternative conditions that lead to the same action are joined with or.
Short-circuit evaluation skips the right operand of and when the left is false, and the right operand of or when the left is true, so ( i < n ) and ( a[i] > 0 ) never reads a[n].

Last updated: September 2026

What this competency asks

Within the operators competency, ETS lists two Boolean skills:

Use Boolean algebra to identify equivalent Boolean expressions.
Write a Boolean expression equivalent to given code, or identify code equivalent to a given Boolean expression or English description.

ETS's sample question on this topic gives ( age > 10 ) and ( height > 36 ) and asks for an equivalent expression. The answer is not ( ( age ≤ 10 ) or ( height ≤ 36 ) ), which uses De Morgan's law and flips each comparison correctly. The trap choices negate > as < instead of ≤, or keep and where De Morgan's law requires or.

Operators and truth tables

A	B	not A	A and B	A or B	A xor B
false	false	true	false	false	false
false	true	true	false	true	true
true	false	false	false	true	true
true	true	false	true	true	false

ETS pseudocode uses the words and, or, and not. Other languages write &&, ||, and !. XOR (exclusive or) is true when exactly one operand is true. It is not an ETS keyword, but it can be written as ( A or B ) and not ( A and B ).

A truth table for n variables has 2ⁿ rows. Two expressions are equivalent exactly when their truth-table columns match in every row. For three variables that is 8 rows. For example, ( A and not B ) or ( not A and C ) is true in rows (A, B, C) = (F, F, T), (F, T, T), (T, F, F), and (T, F, T), and false in the other four.

Precedence

not is evaluated first, then and, then or. So A or B and C means A or ( B and C ), and not A and B means ( not A ) and B. With A = false, B = true, and C = true:

not A and B or not C and A = ( true and true ) or ( false and false ) = true or false = true.

Laws of Boolean algebra

Law	And form	Or form
Identity	A and true = A	A or false = A
Domination	A and false = false	A or true = true
Idempotent	A and A = A	A or A = A
Complement	A and not A = false	A or not A = true
Double negation	not ( not A ) = A
Commutative	A and B = B and A	A or B = B or A
Distributive	A and ( B or C ) = ( A and B ) or ( A and C )	A or ( B and C ) = ( A or B ) and ( A or C )
Absorption	A and ( A or B ) = A	A or ( A and B ) = A
De Morgan	not ( A and B ) = not A or not B	not ( A or B ) = not A and not B

Worked simplification

Simplify not ( not A or B ) or ( A and B ):

De Morgan on the first part: not ( not A or B ) = not not A and not B = A and not B.
The expression is now ( A and not B ) or ( A and B ).
Factor out A: A and ( not B or B ).
Complement law: not B or B = true, so the expression is A and true.
Identity law: A.

Negating comparisons

To apply De Morgan to real conditions, you must negate each comparison exactly:

Condition	Its negation
`x > 10`	`x ≤ 10`
`x ≥ 10`	`x < 10`
`x == y`	`x ≠ y`
`( x ≥ 10 ) and ( x ≤ 20 )` (in the range)	`( x < 10 ) or ( x > 20 )` (outside the range)

Example: the loop while ( ( loggedIn ) and ( tries < 5 ) ) stops when not ( loggedIn and ( tries < 5 ) ) becomes true, which is ( not loggedIn ) or ( tries ≥ 5 ).

From code to a Boolean expression, and back

Nested ifs mean "and"

if ( x > 0 )
    if ( y > 0 )
        print "both positive"
    end if
end if

This prints exactly when ( x > 0 ) and ( y > 0 ).

Separate conditions with the same action mean "or"

boolean discount ← false
if ( age < 13 )
    discount ← true
end if
if ( age ≥ 65 )
    discount ← true
end if

This is equivalent to discount ← ( age < 13 ) or ( age ≥ 65 ).

If/else that returns a Boolean

if ( score ≥ 70 ) return true else return false end if is simply return score ≥ 70.

From English

Description	Expression
"x is between 10 and 20, inclusive"	`( x ≥ 10 ) and ( x ≤ 20 )`
"x is outside that range"	`( x < 10 ) or ( x > 20 )`
"neither a nor b is true"	`not a and not b`, which is `not ( a or b )`
"not both a and b"	`not ( a and b )`, which is `not a or not b`
"exactly one of a and b"	`( a or b ) and not ( a and b )`

A frequent error is writing ( x < 10 ) and ( x > 20 ) for "outside the range." No number is both below 10 and above 20, so that condition is always false.

Short-circuit evaluation

Most languages stop evaluating as soon as the result is known:

In A and B, if A is false, B is not evaluated (the result must be false).
In A or B, if A is true, B is not evaluated (the result must be true).

Programmers use this as a guard:

if ( ( i < n ) and ( list[i] == target ) )

When i = n, the left side is false, so list[n] is never read. Reversing the order would read past the end. The same idea guards against division by zero, as in ( count ≠ 0 ) and ( total / count > 50 ). Because the right side might not run, avoid putting required actions, such as a procedure call that updates a counter, inside a condition.

Boolean logic in hardware

The same algebra describes logic gates. AND, OR, and NOT gates, built from transistors, compute these functions on voltages. NAND (not-and) and NOR (not-or) are each functionally complete: any Boolean function can be built using only NAND gates, or only NOR gates. For example, NOT A = A NAND A. This is why logic gates sit near the bottom of the abstraction hierarchy in Section 4.1.

Test Your Knowledge

Which expression is equivalent to not ( ( temp ≥ 32 ) and ( weather ≠ "snow" ) )?

( temp < 32 ) or ( weather == "snow" )

( temp < 32 ) and ( weather == "snow" )

( temp ≤ 32 ) or ( weather ≠ "snow" )

( temp > 32 ) or ( weather == "snow" )

Test Your Knowledge

Which single condition makes the following code print "both" in exactly the same situations?

if ( x > 0 )
    if ( y > 0 )
        print "both"
    end if
end if

if ( ( x > 0 ) or ( y > 0 ) )

if ( ( x > 0 ) and ( y > 0 ) )

if ( not ( ( x > 0 ) and ( y > 0 ) ) )

if ( ( x ≥ 0 ) and ( y ≥ 0 ) )

Test Your Knowledge

Which condition is true exactly when x is outside the range 10 to 20, inclusive?

( x < 10 ) and ( x > 20 )

( x ≤ 10 ) or ( x ≥ 20 )

not ( ( x < 10 ) or ( x > 20 ) )

( x < 10 ) or ( x > 20 )

Test Your Knowledge

An array list has valid indexes 0 through n − 1. Which condition safely checks whether the element at index i equals target when i might equal n, assuming short-circuit evaluation?

( list[i] == target ) and ( i < n )

( list[i] == target ) or ( i < n )

( i < n ) and ( list[i] == target )

( i < n ) or ( list[i] == target )

Up Next

8.1 Sequence and Selection: if, else, Chained and Nested Conditions

Chapter 8: Programming: Selection, Iteration, and Writing Code

Praxis Computer Science (5652)

7.4 Boolean Expressions, Truth Tables, and Boolean Algebra

Key Takeaways

What this competency asks

Operators and truth tables

Precedence

Laws of Boolean algebra

Worked simplification

Negating comparisons

From code to a Boolean expression, and back

Short-circuit evaluation

Boolean logic in hardware