Question

Problem 2: show all work  

A. Find the complement of F = WX + YZ. B. Show that FF’ = 0 C. Show that F + F’ = 1

Answer 1

The solutions are given below.

A.

Given, F = WX + YZ

Therefore, complement of F i.e. F' can be represented as:

F' = (WX + YZ)'

Apply De Morgan's law i.e. (A + B)' = A'.B'

F' = (WX)'.(YZ)'

Apply De Morgan's law for (WX)' and (YZ)'

F' = (W' + X').(Y' + Z')

Therfore, F' = (W' + X').(Y' + Z').

B.

To prove: F.F' = 0.

Given, F = WX + YZ and F' = (W' + X').(Y' + Z').

Proof:

Let, F.F' = 0.

Substitute the values of F and F'.

(WX + YZ).(W' + X').(Y' + Z') = 0

Expand (W' + X').(Y' + Z').

(WX + YZ).(W'Y' + W'Z' + X'Y' + X'Z') = 0

Expand it completely by multiplying with each other.

(WX.W'Y' + WX.W'Z' + WX.X'Y' + WX.X'Z' + YZ.W'Y' + YZ.W'Z' + YZ.X'Y' + YZ.X'Z') = 0

According to complement law, A.A' = 0 and A + A' = 1. Apply it on the above equation. The terms WW', XX', YY', ZZ' will become equal to zero as per this rule.

(0.XY' + 0.XZ' + 0.WY' + 0.WZ' + 0.ZW' + 0.YW' + 0.ZX' + 0.YX') = 0

According to Annulment rule, A.0 = 0. Apply this on the above equation.

(0 + 0 + 0 + 0 + 0 + 0 + 0 + 0) = 0

In the Left Hand Side, all terms are zero. The operation 0 + 0 will result in 0 itself. Thus, LHS = RHS.

Thus, F.F' = 0 and that concludes the proof.

C.

To prove: F + F' = 1.

Given, F = WX + YZ and F' = (W' + X').(Y' + Z').

Proof:

Let the term WX and YZ be the same as A and B respectively.

So, the equation F can be rewritten as:

F = A + B

Taking the complement of F gives F'. So, after applying the De Morgan's rule, F' becomes:

F' = (A + B)'

F = A'B'

Let, F + F' is equal to 1. Then,

F + F' = 1

Substitute the new values of F and F' to the above equation.

(A+B) + (A'.B') = 1

The term A'.B' can be replaced with (A+B)' by using De Morgan's rule.

(A+B) + (A + B)' = 1

As per the complement rule, A + A' equals 1. So, (A + B) + (A + B)' also equals 1.

Thus, LHS = RHS.

Therefore, F + F' = 1 and that concludes the proof.

Hope this helps. Doubts, if any, can be asked in the comment section.