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Example 6: Prove that (a ∧ (a → b)) → ∼a is a contingency.
Ans.
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. So, 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. NOT gate requires a single input
and is also called a unary gate. It inverts the output. Input 1 is changed to 0 and 0 to 1. NOT gate is represented as a
–
complement (') or bar ( ) at the top.
The symbol and truth table of NOT gate are given below:
A A'
A A'
0 1
1 0
Symbol of NOT gate
Truth table of NOT gate
3.7.2 AND Gate
The AND gate is a logical gate that always gives single output for two and more than two input signals. The AND gate
requires at least two inputs and is called a binary gate. It produces a high output (1) only when all the inputs are
high (1). In all other cases, low output (0) is produced. AND operation is represented by a dot (.).
7878 Touchpad Computer Science-XI

