Essential Logic Gates

Logic gates are the building blocks of digital circuits, representing fundamental operations in Boolean algebra. They process binary inputs to produce outputs, forming the basis for complex logical expressions and decision-making in computer systems. Understanding these gates is key in Discrete Mathematics.

1. AND gate

   • Outputs true (1) only if all inputs are true (1).
   • Symbolized by a multiplication operation (A · B).
   • Fundamental in constructing logical expressions and circuits.
   • Used in decision-making processes where all conditions must be met.
   • Truth table: Only one output (1) for the input combination (1, 1).

2. OR gate

   • Outputs true (1) if at least one input is true (1).
   • Symbolized by an addition operation (A + B).
   • Essential for scenarios where any condition being true is sufficient.
   • Combines multiple signals in digital circuits.
   • Truth table: Outputs (1) for input combinations (1, 0), (0, 1), and (1, 1).

3. NOT gate

   • Outputs the inverse of the input; true (1) becomes false (0) and vice versa.
   • Symbolized by a bar over the variable (¬A or A').
   • Fundamental for negation in logical expressions.
   • Used to create more complex logic circuits by inverting signals.
   • Truth table: Outputs (1) for input (0) and (0) for input (1).

4. NAND gate

   • Outputs false (0) only if all inputs are true (1); otherwise, it outputs true (1).
   • Combination of an AND gate followed by a NOT gate.
   • Universal gate; can be used to create any other logic gate.
   • Important in digital circuit design for simplifying logic.
   • Truth table: Outputs (1) for all combinations except (1, 1).

5. NOR gate

   • Outputs true (1) only if all inputs are false (0); otherwise, it outputs false (0).
   • Combination of an OR gate followed by a NOT gate.
   • Also a universal gate; can be used to construct any other logic gate.
   • Useful in creating circuits that require a default false state.
   • Truth table: Outputs (1) only for input combination (0, 0).

6. XOR gate

   • Outputs true (1) if an odd number of inputs are true (1).
   • Symbolized by A ⊕ B.
   • Commonly used in arithmetic operations and error detection.
   • Represents exclusive conditions where only one input should be true.
   • Truth table: Outputs (1) for input combinations (1, 0) and (0, 1).

7. XNOR gate

   • Outputs true (1) if an even number of inputs are true (1).
   • Combination of an XOR gate followed by a NOT gate.
   • Useful for equality checking in digital circuits.
   • Represents inclusive conditions where inputs must match.
   • Truth table: Outputs (1) for input combinations (0, 0) and (1, 1).