Truth table exercises with answers: 4 variables

Ten formulas over A, B, C and D, sixteen rows each. Build the table, list the TRUE rows and check them against the engine's table under each problem; with sixteen rows, the fastest honest method is to describe the TRUE set by conditions and then count.

A is the most significant bit and D the least, so the rows run from 0, all FALSE, to 15, all TRUE; row 8 is A = TRUE, B = FALSE, C = FALSE, D = FALSE and row 3 is A = FALSE, B = FALSE, C = TRUE, D = TRUE. The solutions describe each TRUE set by the conditions that produce it and name the rows that are easy to miss, such as the row that two groups share and must be counted once.

The functions are the ones that come up in digital design and in rules with four conditions: two ANDs joined by OR, four-input parity, a product of two sums, agreement of two pairs, two implications in conjunction, a three-of-four quorum, a comparison on a binary digit, a double NOR, and an XOR of two ANDs. Counting the FALSE rows is often quicker than counting the TRUE ones; the solutions say which is easier for each.

1. Build the truth table of (A AND B) OR (C AND D) and list the TRUE rows. (A AND B) OR (C AND D)

#	A	B	C	D	f
0	FALSE	FALSE	FALSE	FALSE	FALSE
1	FALSE	FALSE	FALSE	TRUE	FALSE
2	FALSE	FALSE	TRUE	FALSE	FALSE
3	FALSE	FALSE	TRUE	TRUE	TRUE
4	FALSE	TRUE	FALSE	FALSE	FALSE
5	FALSE	TRUE	FALSE	TRUE	FALSE
6	FALSE	TRUE	TRUE	FALSE	FALSE
7	FALSE	TRUE	TRUE	TRUE	TRUE
8	TRUE	FALSE	FALSE	FALSE	FALSE
9	TRUE	FALSE	FALSE	TRUE	FALSE
10	TRUE	FALSE	TRUE	FALSE	FALSE
11	TRUE	FALSE	TRUE	TRUE	TRUE
12	TRUE	TRUE	FALSE	FALSE	TRUE
13	TRUE	TRUE	FALSE	TRUE	TRUE
14	TRUE	TRUE	TRUE	FALSE	TRUE
15	TRUE	TRUE	TRUE	TRUE	TRUE

The engine found no law that shortens this expression; it is already as short as its rewriting laws can make it.

(A AND B) OR (C AND D)

Classification: contingent

Seven rows: the four with A = TRUE, B = TRUE (whatever C and D are), plus the other three with C = TRUE, D = TRUE, namely with A = FALSE, B = FALSE, with A = FALSE, B = TRUE and with A = TRUE, B = FALSE. The row with all four TRUE belongs to both groups and is counted once.

2. Build the truth table of A XOR B XOR C XOR D and list the TRUE rows. A XOR B XOR C XOR D

#	A	B	C	D	f
0	FALSE	FALSE	FALSE	FALSE	FALSE
1	FALSE	FALSE	FALSE	TRUE	TRUE
2	FALSE	FALSE	TRUE	FALSE	TRUE
3	FALSE	FALSE	TRUE	TRUE	FALSE
4	FALSE	TRUE	FALSE	FALSE	TRUE
5	FALSE	TRUE	FALSE	TRUE	FALSE
6	FALSE	TRUE	TRUE	FALSE	FALSE
7	FALSE	TRUE	TRUE	TRUE	TRUE
8	TRUE	FALSE	FALSE	FALSE	TRUE
9	TRUE	FALSE	FALSE	TRUE	FALSE
10	TRUE	FALSE	TRUE	FALSE	FALSE
11	TRUE	FALSE	TRUE	TRUE	TRUE
12	TRUE	TRUE	FALSE	FALSE	FALSE
13	TRUE	TRUE	FALSE	TRUE	TRUE
14	TRUE	TRUE	TRUE	FALSE	TRUE
15	TRUE	TRUE	TRUE	TRUE	FALSE

The engine found no law that shortens this expression; it is already as short as its rewriting laws can make it.

A XOR B XOR C XOR D

Classification: contingent

Parity again: TRUE on the eight rows with an odd number of TRUE inputs, the four with exactly one TRUE input and the four with exactly three. Both the all-FALSE row and the all-TRUE row are FALSE.

3. Build the truth table of (A OR B) AND (C OR D) and list the TRUE rows. (A OR B) AND (C OR D)

#	A	B	C	D	f
0	FALSE	FALSE	FALSE	FALSE	FALSE
1	FALSE	FALSE	FALSE	TRUE	FALSE
2	FALSE	FALSE	TRUE	FALSE	FALSE
3	FALSE	FALSE	TRUE	TRUE	FALSE
4	FALSE	TRUE	FALSE	FALSE	FALSE
5	FALSE	TRUE	FALSE	TRUE	TRUE
6	FALSE	TRUE	TRUE	FALSE	TRUE
7	FALSE	TRUE	TRUE	TRUE	TRUE
8	TRUE	FALSE	FALSE	FALSE	FALSE
9	TRUE	FALSE	FALSE	TRUE	TRUE
10	TRUE	FALSE	TRUE	FALSE	TRUE
11	TRUE	FALSE	TRUE	TRUE	TRUE
12	TRUE	TRUE	FALSE	FALSE	FALSE
13	TRUE	TRUE	FALSE	TRUE	TRUE
14	TRUE	TRUE	TRUE	FALSE	TRUE
15	TRUE	TRUE	TRUE	TRUE	TRUE

The engine found no law that shortens this expression; it is already as short as its rewriting laws can make it.

(A OR B) AND (C OR D)

Classification: contingent

Nine rows: one of A, B must be TRUE and one of C, D must be TRUE. The FALSE rows are the four with A = FALSE, B = FALSE and the four with C = FALSE, D = FALSE, minus the all-FALSE row counted in both: seven in all.

4. Build the truth table of (A <-> B) AND (C <-> D): on which rows do both pairs agree? (A <-> B) AND (C <-> D)

#	A	B	C	D	f
0	FALSE	FALSE	FALSE	FALSE	TRUE
1	FALSE	FALSE	FALSE	TRUE	FALSE
2	FALSE	FALSE	TRUE	FALSE	FALSE
3	FALSE	FALSE	TRUE	TRUE	TRUE
4	FALSE	TRUE	FALSE	FALSE	FALSE
5	FALSE	TRUE	FALSE	TRUE	FALSE
6	FALSE	TRUE	TRUE	FALSE	FALSE
7	FALSE	TRUE	TRUE	TRUE	FALSE
8	TRUE	FALSE	FALSE	FALSE	FALSE
9	TRUE	FALSE	FALSE	TRUE	FALSE
10	TRUE	FALSE	TRUE	FALSE	FALSE
11	TRUE	FALSE	TRUE	TRUE	FALSE
12	TRUE	TRUE	FALSE	FALSE	TRUE
13	TRUE	TRUE	FALSE	TRUE	FALSE
14	TRUE	TRUE	TRUE	FALSE	FALSE
15	TRUE	TRUE	TRUE	TRUE	TRUE

The engine found no law that shortens this expression; it is already as short as its rewriting laws can make it.

(A <-> B) AND (C <-> D)

Classification: contingent

Both pairs must agree, A with B and C with D: four rows, all FALSE; A = FALSE, B = FALSE, C = TRUE, D = TRUE; A = TRUE, B = TRUE, C = FALSE, D = FALSE; all TRUE.

5. Build the truth table of A AND NOT B AND (C OR D) and list the TRUE rows. A AND NOT B AND (C OR D)

#	A	B	C	D	f
0	FALSE	FALSE	FALSE	FALSE	FALSE
1	FALSE	FALSE	FALSE	TRUE	FALSE
2	FALSE	FALSE	TRUE	FALSE	FALSE
3	FALSE	FALSE	TRUE	TRUE	FALSE
4	FALSE	TRUE	FALSE	FALSE	FALSE
5	FALSE	TRUE	FALSE	TRUE	FALSE
6	FALSE	TRUE	TRUE	FALSE	FALSE
7	FALSE	TRUE	TRUE	TRUE	FALSE
8	TRUE	FALSE	FALSE	FALSE	FALSE
9	TRUE	FALSE	FALSE	TRUE	TRUE
10	TRUE	FALSE	TRUE	FALSE	TRUE
11	TRUE	FALSE	TRUE	TRUE	TRUE
12	TRUE	TRUE	FALSE	FALSE	FALSE
13	TRUE	TRUE	FALSE	TRUE	FALSE
14	TRUE	TRUE	TRUE	FALSE	FALSE
15	TRUE	TRUE	TRUE	TRUE	FALSE

The engine found no law that shortens this expression; it is already as short as its rewriting laws can make it.

A AND NOT B AND (C OR D)

Classification: contingent

A must be TRUE, B FALSE and at least one of C, D TRUE: three rows, A = TRUE, B = FALSE with C = FALSE, D = TRUE, with C = TRUE, D = FALSE or with C = TRUE, D = TRUE.

6. Build the truth table of (A -> B) AND (C -> D): which rows survive both implications? (A -> B) AND (C -> D)

#	A	B	C	D	f
0	FALSE	FALSE	FALSE	FALSE	TRUE
1	FALSE	FALSE	FALSE	TRUE	TRUE
2	FALSE	FALSE	TRUE	FALSE	FALSE
3	FALSE	FALSE	TRUE	TRUE	TRUE
4	FALSE	TRUE	FALSE	FALSE	TRUE
5	FALSE	TRUE	FALSE	TRUE	TRUE
6	FALSE	TRUE	TRUE	FALSE	FALSE
7	FALSE	TRUE	TRUE	TRUE	TRUE
8	TRUE	FALSE	FALSE	FALSE	FALSE
9	TRUE	FALSE	FALSE	TRUE	FALSE
10	TRUE	FALSE	TRUE	FALSE	FALSE
11	TRUE	FALSE	TRUE	TRUE	FALSE
12	TRUE	TRUE	FALSE	FALSE	TRUE
13	TRUE	TRUE	FALSE	TRUE	TRUE
14	TRUE	TRUE	TRUE	FALSE	FALSE
15	TRUE	TRUE	TRUE	TRUE	TRUE

The engine found no law that shortens this expression; it is already as short as its rewriting laws can make it.

(A -> B) AND (C -> D)

Classification: contingent

Remove the four rows with A = TRUE, B = FALSE and the four with C = TRUE, D = FALSE; the row A = TRUE, B = FALSE, C = TRUE, D = FALSE is in both lists, so seven rows go and nine remain.

7. Build the truth table of the three-of-four quorum (A AND B AND C) OR (A AND B AND D) OR (A AND C AND D) OR (B AND C AND D) and list the TRUE rows. (A AND B AND C) OR (A AND B AND D) OR (A AND C AND D) OR (B AND C AND D)

#	A	B	C	D	f
0	FALSE	FALSE	FALSE	FALSE	FALSE
1	FALSE	FALSE	FALSE	TRUE	FALSE
2	FALSE	FALSE	TRUE	FALSE	FALSE
3	FALSE	FALSE	TRUE	TRUE	FALSE
4	FALSE	TRUE	FALSE	FALSE	FALSE
5	FALSE	TRUE	FALSE	TRUE	FALSE
6	FALSE	TRUE	TRUE	FALSE	FALSE
7	FALSE	TRUE	TRUE	TRUE	TRUE
8	TRUE	FALSE	FALSE	FALSE	FALSE
9	TRUE	FALSE	FALSE	TRUE	FALSE
10	TRUE	FALSE	TRUE	FALSE	FALSE
11	TRUE	FALSE	TRUE	TRUE	TRUE
12	TRUE	TRUE	FALSE	FALSE	FALSE
13	TRUE	TRUE	FALSE	TRUE	TRUE
14	TRUE	TRUE	TRUE	FALSE	TRUE
15	TRUE	TRUE	TRUE	TRUE	TRUE

The engine found no law that shortens this expression; it is already as short as its rewriting laws can make it.

(A AND B AND C) OR (A AND B AND D) OR (A AND C AND D) OR (B AND C AND D)

Classification: contingent

At least three TRUE inputs: the four rows with exactly one FALSE input and the all-TRUE row, five in all. Each three-input AND term covers one of the first four rows, and all four cover the last.

8. Read A, B, C, D as the bits 8, 4, 2, 1 of a binary digit. Build the table of A OR (B AND (C OR D)) and list the TRUE rows. A OR (B AND (C OR D))

#	A	B	C	D	f
0	FALSE	FALSE	FALSE	FALSE	FALSE
1	FALSE	FALSE	FALSE	TRUE	FALSE
2	FALSE	FALSE	TRUE	FALSE	FALSE
3	FALSE	FALSE	TRUE	TRUE	FALSE
4	FALSE	TRUE	FALSE	FALSE	FALSE
5	FALSE	TRUE	FALSE	TRUE	TRUE
6	FALSE	TRUE	TRUE	FALSE	TRUE
7	FALSE	TRUE	TRUE	TRUE	TRUE
8	TRUE	FALSE	FALSE	FALSE	TRUE
9	TRUE	FALSE	FALSE	TRUE	TRUE
10	TRUE	FALSE	TRUE	FALSE	TRUE
11	TRUE	FALSE	TRUE	TRUE	TRUE
12	TRUE	TRUE	FALSE	FALSE	TRUE
13	TRUE	TRUE	FALSE	TRUE	TRUE
14	TRUE	TRUE	TRUE	FALSE	TRUE
15	TRUE	TRUE	TRUE	TRUE	TRUE

The engine found no law that shortens this expression; it is already as short as its rewriting laws can make it.

A OR (B AND (C OR D))

Classification: contingent

The formula is TRUE when the digit is at least 5. The eight rows with A = TRUE (digits 8 to 15) are TRUE at once, and among the rows with A = FALSE only those with B = TRUE and C or D TRUE remain (digits 5, 6 and 7).

9. Build the truth table of NOR(A, B) AND NOR(C, D) and list the TRUE rows. NOR(A, B) AND NOR(C, D)

#	A	B	C	D	f
0	FALSE	FALSE	FALSE	FALSE	TRUE
1	FALSE	FALSE	FALSE	TRUE	FALSE
2	FALSE	FALSE	TRUE	FALSE	FALSE
3	FALSE	FALSE	TRUE	TRUE	FALSE
4	FALSE	TRUE	FALSE	FALSE	FALSE
5	FALSE	TRUE	FALSE	TRUE	FALSE
6	FALSE	TRUE	TRUE	FALSE	FALSE
7	FALSE	TRUE	TRUE	TRUE	FALSE
8	TRUE	FALSE	FALSE	FALSE	FALSE
9	TRUE	FALSE	FALSE	TRUE	FALSE
10	TRUE	FALSE	TRUE	FALSE	FALSE
11	TRUE	FALSE	TRUE	TRUE	FALSE
12	TRUE	TRUE	FALSE	FALSE	FALSE
13	TRUE	TRUE	FALSE	TRUE	FALSE
14	TRUE	TRUE	TRUE	FALSE	FALSE
15	TRUE	TRUE	TRUE	TRUE	FALSE

The engine found no law that shortens this expression; it is already as short as its rewriting laws can make it.

NOR(A, B) AND NOR(C, D)

Classification: contingent

Each NOR is TRUE only when both of its inputs are FALSE, so the whole formula is TRUE on a single row, all four FALSE. It equals NOT (A OR B OR C OR D).

10. Build the truth table of (A AND B) XOR (C AND D) and list the TRUE rows. (A AND B) XOR (C AND D)

#	A	B	C	D	f
0	FALSE	FALSE	FALSE	FALSE	FALSE
1	FALSE	FALSE	FALSE	TRUE	FALSE
2	FALSE	FALSE	TRUE	FALSE	FALSE
3	FALSE	FALSE	TRUE	TRUE	TRUE
4	FALSE	TRUE	FALSE	FALSE	FALSE
5	FALSE	TRUE	FALSE	TRUE	FALSE
6	FALSE	TRUE	TRUE	FALSE	FALSE
7	FALSE	TRUE	TRUE	TRUE	TRUE
8	TRUE	FALSE	FALSE	FALSE	FALSE
9	TRUE	FALSE	FALSE	TRUE	FALSE
10	TRUE	FALSE	TRUE	FALSE	FALSE
11	TRUE	FALSE	TRUE	TRUE	TRUE
12	TRUE	TRUE	FALSE	FALSE	TRUE
13	TRUE	TRUE	FALSE	TRUE	TRUE
14	TRUE	TRUE	TRUE	FALSE	TRUE
15	TRUE	TRUE	TRUE	TRUE	FALSE

The engine found no law that shortens this expression; it is already as short as its rewriting laws can make it.

(A AND B) XOR (C AND D)

Classification: contingent

Exactly one of the two AND terms must be TRUE: the three rows with A = TRUE, B = TRUE where C and D are not both TRUE, and the three with C = TRUE, D = TRUE where A and B are not both TRUE. The all-TRUE row makes both terms TRUE and is FALSE.

Questions

Is there a shortcut for the XOR chain?

Yes: a chain of XOR is TRUE exactly when an odd number of its inputs is TRUE, so for four inputs you take the four rows with one TRUE input and the four with three. It is not “exactly one TRUE”, and the all-TRUE row is FALSE because four is even.

Guides

How to make a truth table

Tools

Truth table generator: any Boolean expression, up to 12 variables, with intermediate columns

Practice

Generated from the Tangibool engine, semantics tangibool-semantics/1, engine 1.0.0, build 2026-11-07. Every table, map, simplification and verdict on these pages was computed by the Tangibool engine when the site was built, and every claim was re-checked by the same exact checker the app uses. Limits are stated where they apply. © 2026 Foreranger LTD.

Truth table exercises with answers: 4 variables

Questions

Related

Guides

Tools

See also

Practice