Basic logic gates

Logic gates are transistor-based components. Here are the four basic ones.

NOT (complement)

for a variable x, x̄ = 1-x ∴ x + x̄ = 1

OR (union)

OR function (+) has value 1 if any of inputs has value 1.

Properties:

• 1 + x = 1
• 0 + x = x

AND (intersection)

AND function (⋅) has value 1 if all inputs have value 1.

NOTE: the operator ‘⋅’ is often omitted!

Properties:

• x₁ ⋅ x₂ = x₂ ⋅ x₁
• 1 ⋅ x = x

XOR (exclusive or)

XOR function (⨁) has value 1 if one of inputs has value 1.

• x₁ ⨁ x₂ = x₂ ⨁ x₁
• 1 ⨁ x = x̄
• 0 ⨁ x = x

How an XOR gate is built and functions: