The combinational logic circuits are said to be the digital circuits which are made by the combinations of basic logic gates (AND, OR, NOT), they cannot store any memory element, and their output is only depend on the present input stage, the combinational circuits can be of different levels according to their complexities, they can also form by using Universal gates (NAND, NOR gates) to reduce their complexity and to make them simple and complicated.

For each circuit formed the following can be used to represent it combinations:

# 1. Truth Table:

The Truth table is form for defining the maximum combinations of functions generated for the circuit’s outputs.

# 2. Boolean Expression:

Boolean expression is an algebraic representation for any logic circuit or gate which shows its operations in term of individual variables algebraically.

# 3. Logic diagram:

Logic diagram is the graphical representation of a logic circuit or gate which shows the functioning of a logic circuit or gate graphically.

There are some of the commonly known combinational circuits listed below:

**Half Adder****Full Adder****Half Subtractor****Full Subtractor****Multiplexer****Demultiplexer****Encoder****Decoder**

# 1. Half Adder:

A Half Adder is a logical combinational circuit which is used to performs the addition of two 1-bit numbers and there is no provision to add carry, it has two 1- bit inputs and two outputs as Sum and Carry.

**Truth table:**

**Boolean expression:**

Boolean expression for two variables A & B is represented as:

For sum: Sum(S) = A̅B + AB̅ = AꚚB

For carry: Carry(C) = AB

**Circuit diagram:**

# 2. Full Adder:

Unlike half adder which has only two 1- bit inputs and which don’t have any provision to add a carry that could have been generated from lower bit addition, this limitation of half adder is overcome in full adder hence full adder is a combinational logic circuit that has the provision to add a carry.

Let us consider A & B as
two 1- bit inputs C_{in} is a carry generated from the previous order
bit additions, S(Sum) and C(Carry) are the outputs of the full adder.

**Truth table:**

**Boolean expression:**

Boolean expression for three variables A & B and a carry
C_{in} generated in lower bit addition is represented as:

For sum: Sum(S) = AꚚBꚚC_{in}

For carry: Carry(C) = AB + BC_{in} + AC_{in}

**Circuit diagram:**

# 3. Half Subtractor:

A Half Subtractor is a logical combinational circuit which is used to performs the subtraction of two 1-bit numbers. It subtracts B(Subtrahend) from A (Minuend), it has two 1- bit inputs and two outputs as Difference and Borrow.

**Truth table:**

**Boolean expression:**

Boolean expression for two variables A & B is represented as:

For difference: Difference(D) = A̅B + AB̅ = AꚚB

For borrow: Borrow(C) = A̅B

**Circuit diagram:**

# 4. Full Subtractor:

Unlike a half subtractor which has only two 1- bit inputs and which don’t have a provision for subtraction of a borrow that may be generated from lower order bit subtraction, this limitation of half subtractor is overcome in full subtractor hence a full subtractor is a combinational logic circuit that has the provision to take into account a borrow.

**Truth table:**

**Boolean expression:**

Boolean expression for three variables A & B and a borrow
C_{in} generated in lower bit subtraction is represented as:

For difference: Difference(D) = AꚚBꚚC_{in}

For borrow: Borrow(C) = A̅B + BC_{in} + A̅C_{in}

**Circuit diagram: **

# 5. Multiplexer:

The multiplexer is a combinational circuit with two or more (multiple)
inputs and a single output, selectors select one input at a time & send a
to the output line it is also called select inputs or select lines, for n –
input multiplexer m select inputs are required where 2^{m} = n.

**Truth table:**

**Boolean expression:**

Boolean expression of multiplexer according to truth table is represented as:

Y = S̅_{1}S̅_{0} I_{0} + S̅_{1}S_{0} I_{1} + S_{1}S̅_{0}
I_{2} + S_{1}S_{0}
I_{3}

**Circuit diagram: **

# 6. Demultiplexer:

The demultiplexer is a combinational circuit having single input
and two or more (multiple) outputs, it performs just reverse operation of a
multiplexer, it accepts a single input and send it to one of the output lines, for
an n – output demultiplexer, the number of select line required is m where n =
2^{m}.

**Truth table:**

**Boolean expression:**

Boolean expression of demultiplexer according to truth table is represented as:

Y_{0} = S̅_{1}S̅_{0}D

Y_{1 }= S̅_{1}S_{0}D

Y_{2} = S_{1}S̅_{0}D

Y_{3} = S_{1}S_{0}D

**Circuit diagram: **

# 7. Encoder:

An encoder is a combinational circuit, which converts an
active input signal into a coded output signal, it has n – input lines, only
one of which is active at any time and m output lines. Where 2^{m} = n,
in encoder the number of output lines is always less the number of input lines.

**Truth table:**

**Boolean expression:**

Boolean expression of encoder according to table is represented as:

Y_{0}** **= D_{1}
+ D_{3} + D_{5} + D_{7}

Y_{1} = D_{2}
+ D_{3} + D_{6} + D_{7}

Y_{2} = D_{4}
+ D_{5} + D_{6} + D_{7}

**Circuit diagram: **

# 8. Decoder:

The decoder is a combinational circuit that decodes(expand)
the coded input into output signals, it works fully opposite to a encoder, it
convert a n – bit binary input code (data) into 2^{n} output lines such
that each of the output lines will be activated for only one possible
combination of input at a time.

**Truth table:**

**Boolean expression:**

Boolean expression of decoder according to table is represented as:

D_{0 }= D_{0}S_{0}S_{1}

D_{1 }= D_{1}S̅_{0}S_{1}

D_{2 }= D_{2}S_{0}S̅_{1}

D_{3 }= D_{3}S̅_{0}S̅_{1}

**Circuit diagram:**

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