NIMCET 2023 — Mathematics PYQ

Step 1: Write Down the Expression for Negation

We need to find the negation ($\sim$) of the expression:

$$\sim [\sim S\vee (\sim R\wedge S)]$$

Step 2: Apply De Morgan's Law

De Morgan's Laws state that:

1. $\sim (P\vee Q)\equiv \sim P\wedge \sim Q$

2. $\sim (P\wedge Q)\equiv \sim P\vee \sim Q$

Applying the first law to our expression:

$$\sim [\sim S\vee (\sim R\wedge S)]\equiv (\sim \sim S)\wedge \sim (\sim R\wedge S)$$

Using the Double Negation Law ($\sim \sim S\equiv S$):

$$\equiv S\wedge \sim (\sim R\wedge S)$$

Step 3: Apply De Morgan's Law to the Inner Bracket

Now, apply the second De Morgan's Law to the expression inside the second bracket $\sim (\sim R\wedge S)$:

$$\sim (\sim R\wedge S)\equiv (\sim \sim R)\vee (\sim S)$$

$$\equiv R\vee \sim S$$

Substitute this back into the expression from Step 2:

$$\equiv S\wedge (R\vee \sim S)$$

Step 4: Apply the Distributive Law

Now, use the Distributive Law to expand the expression:

$$P\wedge (Q\vee R)\equiv (P\wedge Q)\vee (P\wedge R)$$

Applying this here:

$$\equiv (S\wedge R)\vee (S\wedge \sim S)$$

Step 5: Simplify Using Complement Law

According to the Complement Law, a statement and its negation joined by an "AND" ($\wedge$) operator is always false (a contradiction):

$$S\wedge \sim S\equiv \text{F (False)}$$

Substitute $\text{F}$ back into the equation:

$$\equiv (S\wedge R)\vee \text{F}$$

Any statement joined by an "OR" ($\vee$) operator with False remains the statement itself (Identity Law):

$$\equiv S\wedge R$$

Conclusion

The negation of the statement is simplified to $S\wedge R$.

The correct option is C) S $\wedge$ R.