Page 77 - ComputerScience_Class_11

Example 5: Prove that ∼(a ∨ b) is a contingency.

Answer: a b a ∨ b ∼(a ∨ b)
0 0 0 1
0 1 1 0
1 0 1 0
1 1 1 0
The values in the final column contain both 0 and 1. Hence, it is a contingency.

Example 6: Prove that (a ∧ (a → b)) → ∼a is a contingency.
Answer: a b ∼a a → b a ∧ (a → b) (a ∧ (a → b)) → ∼a
0 0 1 1 0 1
0 1 1 1 0 1
1 0 0 0 0 1
1 1 0 1 1 0

The last column has both 0 and 1 as truth values. Hence, it is a contingency.

3.7 LOGIC GATES
Logic gates are the digital circuits that depict a logical relationship between the input and output voltages of the
circuits. They are the building blocks of a digital circuit.

LOGIC AND OR XOR NOT
GATES

NAND NOR XNOR

A physical gate acts as a barrier and controls entry into the building. Similarly, logic gates are used to control the flow
of information based on the logical relations. A logic gate can accept one or more inputs but always produces a single
output. They produce signal 1 as an output when the input logic requirement is satisfied, otherwise signal 0.

Logic gates require a power supply. In both input and output, 0 volt represents OFF (0) and 1 volt represents ON (1).
Each logic gate is depicted with a specific graphical symbol.

The three basic logical operations namely conjunction (AND), disjunction (OR) and negation (NOT) are represented by
their corresponding logic gates discussed below. These gates are also called the fundamental gates.

3.7.1 NOT Gate
The NOT gate is a logical gate that always gives the opposite output of the input signal. The NOT gate requires a single
input and is also called a unary gate. It inverts the output: Input 1 is changed to 0 input 0 is changed to 1. The NOT gate
–
is represented as a complement (') or a bar ( ) at the top.

Propositional Logic, Hardware Implementation, Arithmetic Operations 75