Boolean Logic, Gates & Truth Tables

What is Boolean logic?

Boolean logic (named after George Boole) works with only two values: True (1) and False (0). A Boolean expression combines these values using operations like NOT, AND and OR. Every decision inside a computer — "is the password correct?", "is marks ≥ 33?" — finally becomes Boolean logic in electronic circuits called logic gates.

Symbols used: NOT A is written A' or Ā; A AND B is A·B; A OR B is A + B; A XOR B is A ⊕ B.

Logic gates at a glance

NOT, AND, OR

NOT gate (inverter)

One input; the output is the opposite. Y = A'.

AND gate

Output is 1 only when all inputs are 1. Y = A·B. Think: "pass only if theory AND practical are cleared".

OR gate

Output is 1 when at least one input is 1. Y = A + B. Think: "entry allowed with ID card OR fee receipt".

NAND and NOR (universal gates)

NAND = AND followed by NOT: Y = (A·B)'. Output is 0 only when both inputs are 1. NOR = OR followed by NOT: Y = (A + B)'. Output is 1 only when both inputs are 0.

They are universal: for example, connecting both inputs of a NAND gate together gives a NOT gate, because (A·A)' = A'. Chips are cheaper to make when only one kind of gate is used.

XOR (exclusive OR)

Output is 1 when the inputs are different. Y = A ⊕ B = A'B + AB'. Example: a staircase light with one switch at the bottom and one at the top — flipping either switch changes the light.

All truth tables with Python

In Python, the bitwise operators & (AND), | (OR) and ^ (XOR) work on 0 and 1, so we can print every table in a few lines:

print('A B | AND OR XOR NAND NOR')
for A in (0, 1):
    for B in (0, 1):
        AND = A & B
        OR = A | B
        XOR = A ^ B
        print(A, B, '|', AND, '  ', OR, ' ', XOR, '  ', 1 - AND, '   ', 1 - OR)

Boolean laws

De Morgan’s laws

Proof by truth table — check that both sides are equal for every row. Python can do all rows for us:

# Check De Morgan's laws for every combination
ok = True
for A in (False, True):
    for B in (False, True):
        law1 = (not (A and B)) == ((not A) or (not B))
        law2 = (not (A or B)) == ((not A) and (not B))
        print(A, B, law1, law2)
        ok = ok and law1 and law2
print('Both laws hold:', ok)

Logic circuits and expressions

A logic circuit joins gates together. To find its truth table, write the output expression and evaluate it for every input combination. Example: Y = (A·B) + C' — one AND gate, one NOT gate and one OR gate.

# Y = (A AND B) OR (NOT C)
print('A B C | Y')
for A in (0, 1):
    for B in (0, 1):
        for C in (0, 1):
            Y = (A and B) or (not C)
            print(A, B, C, '|', int(Y))

Three inputs → 2³ = 8 rows. Y is 1 whenever C is 0, and also when A and B are both 1.

Summary & practice

• Boolean values: 0 and 1. Operations: NOT (A'), AND (A·B), OR (A+B), XOR (A⊕B).
• NAND and NOR are universal gates.
• n inputs → 2ⁿ rows in the truth table.
• De Morgan: (A·B)' = A' + B', (A+B)' = A'·B'.