Simplified form of the Boolean function

is

The correct option is:

$\overline{P}S+\overline{Q}\overline{S}$

Here is the step-by-step simplification of the Boolean function:

Step 1: Write down the original Boolean function
The given Boolean function is:

$F(P,Q,R,S)=\overline{P}\overline{Q}+\overline{P}QS+P\overline{Q}\overline{R}\overline{S}+P\overline{Q}R\overline{S}$

Step 2: Group and simplify the last two terms
We can factor out the common terms from the third and fourth terms:

$P\overline{Q}\overline{R}\overline{S}+P\overline{Q}R\overline{S}=P\overline{Q}\overline{S}(\overline{R}+R)$

Using the Boolean identity $\overline{R}+R=1$, this simplifies to:

$P\overline{Q}\overline{S}$

Substituting this back into our original equation, we get:

$F=\overline{P}\overline{Q}+\overline{P}QS+P\overline{Q}\overline{S}$

Step 3: Expand terms to include missing variables (Shannon's expansion / Consensus theorem method)
We can expand the terms to include the variables $S$ and $\overline{S}$ to facilitate grouping.
Specifically, let's expand the first term $\overline{P}\overline{Q}$ by multiplying it with $(S+\overline{S})$:

$\overline{P}\overline{Q}=\overline{P}\overline{Q}(S+\overline{S})=\overline{P}\overline{Q}S+\overline{P}\overline{Q}\overline{S}$

Now substitute this expanded term back into the function:

$F=\overline{P}\overline{Q}S+\overline{P}\overline{Q}\overline{S}+\overline{P}QS+P\overline{Q}\overline{S}$

Step 4: Group terms with common factors
Group the terms by the variables $S$ and $\overline{S}$:

$F=(\overline{P}\overline{Q}S+\overline{P}QS)+(\overline{P}\overline{Q}\overline{S}+P\overline{Q}\overline{S})$

Factor out the common terms from each grouped set:

$F=\overline{P}S(\overline{Q}+Q)+\overline{Q}\overline{S}(\overline{P}+P)$

Step 5: Apply Boolean properties to simplify
Using the Boolean identity $A+\overline{A}=1$, we have:
1) $\overline{Q}+Q=1$
2) $\overline{P}+P=1$

Substituting these back into the expression:

$F=\overline{P}S(1)+\overline{Q}\overline{S}(1)$

Which simplifies to the final expression:

$F=\overline{P}S+\overline{Q}\overline{S}$