3 input EX-NOR Gate and Its Associativity - Digital Logic

Question

Does EX-NOR Gate follow Associativity ?

How 3 input EX-NOR function is equal to 3 input EX-OR function ?

Comments

3-input XOR function is SAME as 3-input XNOR function.

BUT

3-input XNOR Gate is NOT same 3-input XOR Gate.

Watch this: XNOR Gate Vs XNOR Function | 3 Inputs XNOR gate | GATE EC 2010, GATE EC 2015

Answer 1

 

Yes, EX-NOR Gate follows Associativity. 

 

Associativity Definition: Three elements A, B, C in a non-empty set G are Associative if

$(A \cdot B)\cdot C= A\cdot (B\cdot C)$

(Where $\cdot$ is any binary operator) 

 

Here is some $Standard \ Source$ for more details about $Associative \ Property$ – 

1. Wikipedia
2. Introduction to Group Theory, MIT
3. Introduction to the Theory of Finite Groups, Stanford University

 

XNOR Gate: If both of the given inputs are equal, it gives output 1. It is also called as Equality Detector,

(Source: M. Morris Mano)

Now, let us draw the truth table:

 

A	B	C	(A ⊙ B) ⊙ C	A ⊙ (B ⊙ C)	A ⊙ B ⊙ C	A ⊕ B ⊕ C
0	0	0	0	0	0	0
0	0	1	1	1	1	1
0	1	0	1	1	1	1
0	1	1	0	0	0	0
1	0	0	1	1	1	1
1	0	1	0	0	0	0
1	1	0	0	0	0	0
1	1	1	1	1	1	1

 

So from the above truth table we can observe that,

 

(A ⊙ B) ⊙ C ≡ A ⊙ (B ⊙ C) ≡ A ⊙ B ⊙ C ≡ A ⊕ B ⊕ C ≡ Σm (1,2,4,7)

 

 

Hence, (A ⊙ B) ⊙ C = A ⊙ (B ⊙ C), i.e EX-NOR follows Associativity.

 

 

Some additional points to remember :

 

• For Multiple Inputs:

 

 

$EX-OR \Rightarrow {Detects}\ {odd}\ {no}\ {of}\ {1's}$

$EX-NOR \Rightarrow {Detects}\ {even}\ {no}\ {of}\ {0's}$

 

 

• For Odd No of Inputs:

 

$EX-NOR = {EX-OR}$

 

Because, for odd no of inputs, Odd no of 1’s means Even no of 0’s and vice versa. (By intuition)

Let say, Inputs 001 → Here No of 0’s = 2 (even) and No of 1’s = 1(odd)

 

• For Even No of Inputs:

 

$EX-NOR = \overline{EX-OR}$

 

Truth Table: 

A	B	A ⊙ B	A ⊕ B
0	0	1	0
0	1	0	1
1	0	0	1
1	1	1	0

 

 

• GATE VS FUNCTION:

 

 

3-input EXOR Function is $SAME$ as 3-input EXNOR Function.

 

But, 3-input EXOR Gate is $NOT$ same 3-input EXNOR Gate.

 

 

 

• The Implementation of 3 i/p EX-NOR Gate:

 

 

This is the way to implement 3-i/p XNOR $Gate$, which is equivalent to $complement$ of 3-i/p XOR gate i.e 3 i/p XOR + NOT.

 

 

A ⊙ B ⊙ C =  A ⊕ B ⊕ C   [For Function]

 

A ⊙ B ⊙ C =  $\overline{A \oplus B \oplus C}$  [For Gate]

 

 

Good Read: GO TS - 1 question on 3 input XNOR

Video Explanation: XNOR Gate Vs XNOR Function | 3 Inputs XNOR gate

Comments

Excellent answer

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Excellent Answer @AniMan_7.

Particularly the last point that you mentioned about 3-input XNOR Gate. 

3-input XOR function is SAME as 3-input XNOR function.

BUT

3-input XNOR Gate is NOT same 3-input XOR Gate.

Explained Here: XNOR Gate Vs XNOR Function | 3 Inputs XNOR gate | GATE EC 2010, GATE EC 2015 

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Thank you @Deepak Poonia sir.

Just added gate vs function point too.