If the function X (P, Q, R, S) and Y (P, Q, R, S) are

X = R + P’ Q + R’ S and Y = P’ Q’ R’ S’ + P Q R’ S’ P Q’ R’ S’

(where ’ represent complement). Which of the following is true ? 

Explanation:

Given the functions:

X = R + P’Q + R’S

Y = P’Q’R’S’ + PQR’S’ + PQ’R’S’

Where ‘ represents the complement.

To determine the relationship between X and Y, we first need to simplify and analyze the given expressions.

Step-by-Step Simplification:

1. Simplify the expression for Y:

Y = P’Q’R’S’ + PQR’S’ + PQ’R’S’

First, observe that R’S’ is a common factor in all three terms:

Y = R’S’(P’Q’ + PQ + PQ’)

Next, simplify the expression inside the parentheses:

P’Q’ + PQ + PQ’

Here, we can further analyze the expression:

- P’Q’ remains as is.

- PQ remains as is.

- PQ’ remains as is.

Now, let's check for any simplifications:

P’Q’ + PQ + PQ’

This expression covers all possible combinations of P and Q, except for the case when P and Q are both 1 (which is covered by PQ). Therefore, the expression simplifies to 1 (since it accounts for all possible combinations of P and Q):

P’Q’ + PQ + PQ’ = 1

So, we have:

Y = R’S’(1)

Y = R’S’

2. Compare X and Y:

Now, we have:

X = R + P’Q + R’S

Y = R’S’

To see if X equals Y, let’s evaluate if X can be simplified to R’S’:

X = R + P’Q + R’S

Consider the case when R’S’ is present:

If R = 0 and S = 0, R’S’ = 1.

When R = 0 and S = 0, X = 0 + P’Q + 0 = P’Q.

For X to be equal to R’S’ in all cases, X must simplify to 0 when R = 1 or S = 1:

- If R = 1, X = 1 + anything = 1.

- If S = 1, X = R + P’Q + 1 = 1 + anything = 1.

Therefore, X does not simplify directly to R’S’.

Instead, we need to evaluate if X and Y are complements of each other:

Check X’ (complement of X):

X = R + P’Q + R’S

X’ = (R + P’Q + R’S)’

Using De Morgan’s laws:

X’ = R’P’Q’(R’S)’

X’ = R’P’Q’ + RS’

This does not simplify directly to Y (R’S’).

Conclusion:

Based on our analysis, there is no direct relationship between X and Y, as X does not simplify to Y, and X’ does not simplify to Y either. Therefore, the correct option is:

Option 3: no direct relation between X and Y.

Important Information:

It is crucial to thoroughly analyze and simplify Boolean expressions using appropriate laws and principles. The complement and simplification of complex expressions should be done methodically to avoid errors. Understanding the principles of Boolean algebra is essential for solving such problems accurately.