intermediate Digital Electronics

Truth Tables & Karnaugh Maps

Use the truth table generator to build and simplify Boolean functions. Explore multi-variable expressions and Sum of Products minimisation.

Boolean Expression

Operators: AND OR NOT XOR NAND NOR XNOR — e.g. (A AND B) OR NOT C

Sum of Products (SOP)

NOT A AND NOT B AND NOT C
  OR  NOT A AND B AND NOT C
  OR  A AND NOT B AND NOT C
  OR  A AND B AND NOT C
  OR  A AND B AND C

Enter the 4-variable majority function: '(A AND B) OR (A AND C) OR (A AND D) OR (B AND C) OR (B AND D) OR (C AND D)'. Output is 1 when 2 or more of A,B,C,D are 1.

Show hint

A majority gate is used in fault-tolerant systems (triple modular redundancy) and voting circuits.
Count the rows where OUT = 1. For a 4-input majority function you should have 11 minterms (C(4,2) + C(4,3) + C(4,4) = 6 + 4 + 1 = 11).

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With 4 variables there are 2^4 = 16 rows total. Out of these, 11 have output 1.
Now try the XOR chain: 'A XOR B XOR C'. This is a 3-input parity function — output is 1 when an odd number of inputs are high.

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XOR chains are used in CRC (cyclic redundancy check) circuits and binary adders. The carry-out of a half-adder is A AND B; the sum bit is A XOR B.
Try 'A XNOR B' (equivalence gate). XNOR is 1 when inputs are EQUAL. This is the complement of XOR.

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XNOR is used in comparators. A 4-bit equality comparator is built from four XNOR gates ANDed together.
Enter 'A NAND B'. NAND is functionally complete — any Boolean function can be built using only NAND gates. Verify that output = NOT (A AND B).

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NAND and NOR are universal gates. CMOS logic naturally inverts, so NAND gates are the most efficient in silicon.