Consider the statement "The match will be played only if the weather is good and ground is not wet". Select the correct negation from the following:

Let us translate every part of the given English sentence into symbolic logic so that each subsequent algebraic-style step is absolutely clear.

First, we introduce three simple statements:

$$P : \text{"The match will be played"}$$

$$Q : \text{"The weather is good"}$$

$$R : \text{"The ground is not wet"}$$

The original statement reads: “The match will be played only if the weather is good and the ground is not wet.” In propositional logic, the phrase “only if” is expressed by the implication arrow $$\rightarrow$$, with the condition coming after the arrow. Hence we write

$$P \;\rightarrow\; (Q \land R).$$

Our task is to find the negation of this entire implication. We start by recalling the standard logical equivalence for the negation of an implication:

Formula to be used: $$\neg(A \rightarrow B) \equiv A \land \neg B.$$

Here $$A$$ is $$P$$ and $$B$$ is $$(Q \land R)$$. Applying the formula gives

$$\neg\bigl(P \rightarrow (Q \land R)\bigr) \;=\; P \land \neg(Q \land R).$$

Now we still have the negation of a conjunction inside. We therefore invoke De Morgan’s law, which states

Formula to be used: $$\neg(X \land Y) \equiv \neg X \;\lor\; \neg Y.$$

Taking $$X = Q$$ and $$Y = R$$, De Morgan’s law yields

$$\neg(Q \land R) \;=\; \neg Q \;\lor\; \neg R.$$

Substituting this back into our earlier result, we obtain

$$\neg\bigl(P \rightarrow (Q \land R)\bigr) \;=\; P \land (\neg Q \lor \neg R).$$

Let us now translate this final symbolic form back into ordinary language:

• $$P$$ is true  ⇒  “The match will be played.”
• $$\neg Q$$ is true  ⇒  “The weather is not good.”
• $$\neg R$$ is true  ⇒  “The ground is wet.”

So the entire expression $$P \land (\neg Q \lor \neg R)$$ becomes

“The match will be played and (the weather is not good or the ground is wet).”

Scanning the given options, we see that Option C states exactly this sentence:

“The match will be played and weather is not good or ground is wet.”

Hence, the correct answer is Option C.