Discrete Mathematics Logic Gates and Circuits

Discrete Mathematics Logic Gates and Circuits with introduction, sets theory, types of sets, set operations, algebra of sets, multisets, induction, relations, functions and algorithms etc.

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Logic Gates and Circuits:

Schematic or Graphical Representation of Boolean Expressions:

The Boolean expressions can be graphically represented by using logic circuits. These logic circuits can be constructed using solid-state devices called gates, which are capable of switching voltage levels. If x and y are variables, then the basic expressions x ∧ y (AND), x ∨ y (OR) and x' (NOT) are shown graphically as follows:

AND Gate: An AND gate receives inputs x and y and produces output denoted x ∧ y, as shown in logic table

x	y	x∧ y
0	0	0
0	1	0
1	0	0
1	1	1

OR Gate: An OR gate receives inputs x and y and produces output denoted x V y as shown in the logic table

x	y	x∧ y
0	0	0
0	1	1
1	0	1
1	1	1

NOT Gate: A NOT Gate receives input x and produces output y denoted x' as shown in the logic table

x	x'
0	1
1	0

We can interconnect these devices to form an electronic circuit that realizes any given Boolean Expression.

Applications of Boolean Algebra in Switching Theory:

The simplest switching device is NO-OFF switch. A switch is a device is an electric circuit which lets or does not let the current to flow through the circuit.

The switch has two states NO or OFF (closed or open)

If the switch is closed, current will pass through it.
If the switch is open, current will not pass through it.
ON is denoted by 1

OFF is indicated by 0.

The electrical circuits containing switches by Boolean expressions, if ON is denoted by true or 1 and OFF is denoted by False or 0.

Series and Parallel Connection:

There are 2 ways in which switches are connected with each other.

1. Series Connection 2. Parallel Connection

1. Series Connection: Two switches x¹ and x²are said to be connected in series if current can pass only when both the switches are ON (closed),and the current does not flow if both the switches or any one are OFF (open). Symbolically, series connection of two switches x¹ and x² is denoted by x¹∧ x²or x¹* x².

It is represented diagrammatically as follows:

2. Parallel Connection: Two switches x¹ and x²are said to be connected in parallel if current flows when both any one of the switches are ON (closed) and the current does not flow when both are OFF (open). Symbolically, Parallel connection of two switches x¹ and x² is denoted by x¹∨ x²or x¹+ x².

It is represented diagrammatically as follows:

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