Draw the logic diagram of EX-NOR gate using only NOR gates.

To draw the logic diagram of an EX-NOR (Exclusive-NOR) gate using only NOR gates, we need to understand the logic function of an EX-NOR gate and how it can be constructed using NOR gates. An EX-NOR gate is a digital logic gate that outputs true or 1 only when the number of true inputs is even. The truth table for a 2-input EX-NOR gate is as follows:

A	B	A EX-NOR B
0	0	1
0	1	0
1	0	0
1	1	1

The Boolean expression for a 2-input EX-NOR gate is:

[ A \oplus B = \overline{A \cdot B} + \overline{\overline{A} \cdot \overline{B}} ]

Now, let's construct this using NOR gates. A NOR gate is a universal gate, which means that any other gate can be constructed using only NOR gates. The truth table for a NOR gate is:

A	B	A NOR B
0	0	1
0	1	0
1	0	0
1	1	0

The Boolean expression for a NOR gate is:

[ A \downarrow B = \overline{A + B} ]

To create an EX-NOR gate, we need to implement the EX-NOR Boolean expression using NOR operations. We can break down the EX-NOR expression into simpler parts that can be constructed using NOR gates. The expression can be rewritten using De Morgan's laws and the property of double negation:

[ A \oplus B = \overline{A \cdot B} + \overline{\overline{A} \cdot \overline{B}} = \overline{\overline{\overline{A \cdot B}} \cdot \overline{\overline{\overline{A} \cdot \overline{B}}}} ]

Now, let's construct the EX-NOR gate step by step using NOR gates:

1. Create the term ( A \cdot B ) using NOR gates:

   • ( A \downarrow A = \overline{A + A} = \overline{A} )
   • ( B \downarrow B = \overline{B + B} = \overline{B} )
   • ( (A \downarrow A) \downarrow (B \downarrow B) = \overline{\overline{A} + \overline{B}} = A \cdot B )

2. Create the term ( \overline{A} \cdot \overline{B} ) using NOR gates (similar to step 1).

3. Create the term ( \overline{A \cdot B} ) by NOR-ing the result of step 1 with itself.

4. Create the term ( \overline{\overline{A} \cdot \overline{B}} ) by NOR-ing the result of step 2 with itself.

5. Finally, NOR the results of steps 3 and 4 to get the EX-NOR output.

Here is the logic diagram for the EX-NOR gate using NOR gates:

 A ----|‾‾‾|----|‾‾‾|----|‾‾‾|----|‾‾‾|
       |NOR|    |NOR|    |NOR|    |NOR|---- A EX-NOR B
 B ----|___|----|___|    |___|    |___|
         |        |        |        |
         |--------|        |--------|

In this diagram, the first two NOR gates at the top and bottom create the inverted inputs ( \overline{A} ) and ( \overline{B} ). The next two NOR gates generate ( A \cdot B ) and ( \overline{A} \cdot \overline{B} ). The last NOR gate combines these to produce the EX-NOR output.