Truth Tables Examples in CS Thinking

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Truth Tables.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in CS Thinking.

Concept Recap

A table listing every combination of boolean inputs and the resulting output for a logical expression.

Map out every possible True/False scenario to be sure you understand what a logical expression does.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Truth tables exhaustively verify logical expressions and can reveal equivalences between different expressions.

Common stuck point: With n variables there are 2ⁿ rows — 3 variables means 8 rows, 4 means 16.

Sense of Study hint: To build a truth table, list all input variables as columns, then add a column for the output expression. Fill in all possible True/False combinations for the inputs ( $2^n$ rows for $n$ variables), then evaluate the expression for each row. Compare columns to check for logical equivalences.

Worked Examples

Example 1

medium

Build the truth table for

(A \text{ AND } B) \text{ OR } C

and count the number of True rows.

Answer

5

First step

List 8 rows for

(A,B,C)

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Example 2

medium

Using De Morgan, what expression is equivalent to

\text{NOT}(A \text{ OR } B)

? Then verify on the

A=T,B=F

row.

Example 3

medium

Use a truth table to decide whether

A \text{ XOR } A

is equivalent to False (a contradiction).

A XOR A is always False — a contradiction

Example 4

hard

Build the truth table for

(A \text{ AND } B) \text{ OR } (\text{NOT } A \text{ AND NOT } B)

. What standard operator does it match?

(A AND B) OR (NOT A AND NOT B) — True when A equals B (XNOR)

Example 5

hard

Use a truth table to test whether

A \text{ AND (NOT } A \text{ OR } B)

simplifies to

A \text{ AND } B

Example 6

challenge

Verify with truth tables that NAND alone is functionally complete by expressing NOT, AND, and OR using only NAND.

Example 7

challenge

Two students disagree about whether

(A \text{ OR } B) \text{ AND } (\text{NOT } A \text{ OR } C)

equals

(A \text{ AND } C) \text{ OR } (\text{NOT } A \text{ AND } B)

. Settle it with a truth table.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

How many rows does a truth table for 2 boolean variables have?

Example 2

easy

How many rows does a truth table for 3 boolean variables have?

Example 3

easy

In the truth table for

A \text{ AND } B

, how many of the 4 rows are True?

Truth table for A AND B — only 1 of 4 rows is True

Example 4

easy

In the truth table for

A \text{ OR } B

, how many of the 4 rows are True?

Truth table for A OR B — 3 of 4 rows are True

Example 5

easy

In the truth table for

A \text{ XOR } B

, how many of the 4 rows are True?

Truth table for A XOR B — exactly 2 of 4 rows are True

Example 6

easy

For the row

A=\text{True}, B=\text{False}

, what is

A \text{ AND } B

Fill in: what is A AND B when A=True, B=False?

Example 7

easy

How many rows does a truth table for 4 boolean variables have?

Example 8

easy

For the row

A=\text{False}

, what is

\text{NOT } A

Fill in: NOT A when A=False

Example 9

medium

Build the truth table for

\text{NOT } A \text{ OR } B

. For which of the 4 (A,B) rows is it False?

Which row of NOT A OR B is False?

Example 10

medium

Are

A \text{ AND } B

and

B \text{ AND } A

equivalent? Justify using the truth table.

A AND B vs B AND A — all rows match (commutative)

Example 11

medium

Verify De Morgan: do

\text{NOT(A AND B)}

and

\text{(NOT A) OR (NOT B)}

match on all 4 rows?

De Morgan: NOT(A AND B) matches (NOT A) OR (NOT B) on all 4 rows

Example 12

medium

Evaluate

A \text{ AND (B OR C)}

for

A=\text{True}, B=\text{False}, C=\text{True}

Evaluate A AND (B OR C) for A=T, B=F, C=T step by step

Example 13

medium

In an 8-row truth table for

A,B,C

, how many rows have exactly one variable True?

Example 14

medium

Does

A \text{ OR (A AND B)}

simplify to

A

? Check with the truth table.

Absorption law: A OR (A AND B) matches A on all 4 rows

Example 15

medium

A truth table for an expression in 5 variables would have how many rows, and why is exhaustive checking still feasible by computer?

Example 16

challenge

Two students claim

A \text{ XOR } B

equals

(A \text{ OR } B) \text{ AND NOT}(A \text{ AND } B)

. Verify across ALL 4 rows.

Verify XOR = (A OR B) AND NOT(A AND B) — all 4 rows match

Example 17

challenge

An expression's truth table is True on exactly the rows (T,F) and (F,T) for variables (A,B). Identify the standard operator it represents and write it using AND/OR/NOT.

Recover the operator: True exactly on the two rows where A and B differ

Example 18

challenge

How many distinct truth tables (boolean functions) exist for 2 input variables, and why?

Example 19

medium

Build the truth table for

A \text{ AND NOT } B

. On which (A,B) row is it True?

Fill in: on which row is A AND NOT B True?

Example 20

medium

Are

A \text{ OR } B

and

B \text{ OR } A

equivalent? Justify with the truth table.

A OR B vs B OR A — all rows match (commutative)

Example 21

easy

How many rows does a truth table for

5

boolean variables have?

Example 22

easy

For

A=\text{True}, B=\text{True}

, what is

A \text{ OR } B

Read off: A=True, B=True gives A OR B = True

Example 23

easy

For

A=\text{False}, B=\text{True}

, what is

A \text{ XOR } B

Read off: A=False, B=True gives A XOR B = True

Example 24

easy

How many rows of the 4-row truth table for

A \text{ XOR } B

are False?

XOR truth table — 2 False rows (F,F) and (T,T)

Example 25

easy

Evaluate

\text{NOT}(\text{NOT } A)

when

A=\text{True}

Example 26

easy

Which row of the truth table for

A \text{ AND } B

is the only True row?

AND truth table — the single True row is A=T, B=T

Example 27

easy

True or False: a truth table guarantees logical equivalence only if matching outputs appear on every row.

Example 28

medium

Evaluate

A \text{ OR (NOT } B \text{)}

when

A=\text{False}, B=\text{True}

Evaluate A OR (NOT B) when A=False, B=True

Example 29

medium

Verify whether

A \text{ AND (A OR B)}

is equivalent to

A

using all 4 rows.

Absorption law: A AND (A OR B) equals A on all 4 rows

Example 30

medium

Evaluate

\text{NOT}(A \text{ OR } B)

when

A=\text{False}, B=\text{False}

Evaluate NOT(A OR B) when A=False, B=False

Example 31

medium

In an 8-row truth table for

A,B,C

, how many rows have all three variables True?

Example 32

medium

Evaluate

(A \text{ XOR } B) \text{ AND } C

for

A=T,B=F,C=T

Evaluate (A XOR B) AND C for A=T, B=F, C=T

Example 33

medium

How many True rows does the truth table for

A \text{ OR } B \text{ OR } C

have?

A OR B OR C — 7 of 8 rows are True

Example 34

medium

How many True rows does the truth table for

A \text{ AND } B \text{ AND } C

have?

A AND B AND C — only 1 of 8 rows is True

Example 35

hard

Are

A \text{ AND (B OR C)}

and

(A \text{ AND } B) \text{ OR } (A \text{ AND } C)

equivalent? Justify on all 8 rows.

Example 36

hard

An expression on

A,B

outputs True on rows

(T,T)

and

(F,F)

, False elsewhere. Write a simple AND/OR/NOT expression matching it.

Build the expression that matches this truth table (XNOR pattern)

Example 37

hard

How many distinct boolean functions exist on

3

variables? Justify the count.

Example 38

hard

For

n=10

boolean inputs, exhaustive truth-table checking requires how many rows? Express as a power of two and a decimal.

Example 39

challenge

A 3-input majority function MAJ

(A,B,C)

outputs True when at least two inputs are True. How many True rows does its truth table have, and write it in DNF.

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Related Concepts

Boolean Logic Logical Operators

Background Knowledge

These ideas may be useful before you work through the harder examples.

boolean logiclogical operators