Consider the statement "The match will be played only if the weather is good and ground is not wet".

Q: Wait, why is 'only if' an implication? Shouldn't it be 'if'?

That is a classic trap! In logic, 'p if q' means $q \Rightarrow p$, but 'p only if q' means $p \Rightarrow q$. Think of it this way: 'The match is played' is the result, and 'weather is good AND ground is not wet' are the necessary conditions. If the match is played, those conditions MUST have been met. Hence, $p \Rightarrow (q \wedge r)$.

Q: Why does the negation of an implication not contain an implication arrow?

This is the beauty of logical equivalence! When we say $\sim(p \Rightarrow Q)$, we are saying it is NOT the case that 'if p, then Q'. The only scenario where that statement is false is if p occurs but Q does not. Therefore, the negation is $p \wedge \sim Q$. It is a fundamental rule that simplifies the expression into a conjunction.

Q: Why does the AND symbol flip to an OR symbol when I apply the negation?

That is the essence of De Morgan's Law! Imagine you have two conditions that must both be true for a result. If you want to negate that, you are saying that it is NOT the case that both are true. This means at least one of them must be false. Therefore, the AND ($\wedge$) must become an OR ($\vee$) to represent that either the first condition fails, or the second condition fails, or both.

Question

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:

Options:

(A) The match will not be played and weather is not good and ground is wet.

(B) If the match will not be played, then either weather is not good or ground is wet.

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

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

Sigma Percentile · Accepted Answer

### Analyzing the Setup

To solve this problem, we first define our atomic propositions to transform the English sentence into a precise symbolic structure. We assign the following variables:

$p$: "The match will be played"

$q$: "Weather is good"

$r$: "Ground is not wet"

By breaking the sentence into these components, we transform a vague statement into a rigid mathematical expression.

### The 'Only If' Gatekeeper

We must address the logical connector "only if." In propositional logic, the statement "$p$ only if $Q$" is a conditional statement represented as $p \Rightarrow Q$.

This implies that $p$ cannot occur unless $Q$ is true. Since our condition $Q$ is a compound statement consisting of "weather is good" AND "ground is not wet," we represent it as $(q \wedge r)$.

Thus, our logical foundation is:

$$p \Rightarrow (q \wedge r)$$

### The Art of the Broken Promise

The objective is to find the negation of this statement, denoted as $\sim(p \Rightarrow (q \wedge r))$.

To negate an implication $A \Rightarrow B$, we use the "broken promise" analogy: the only way to invalidate the implication is if $A$ occurs, but $B$ does not. Therefore, the negation of $A \Rightarrow B$ is $A \wedge \sim B$.

Applying this to our expression, where $A = p$ and $B = (q \wedge r)$, we obtain:

$$p \wedge \sim(q \wedge r)$$

### The Final Distribution: De Morgan's Law

We now apply De Morgan's Law to distribute the negation inside the parentheses. De Morgan's Law states that the negation of a conjunction is the disjunction of the negations:

$$\sim(q \wedge r) \equiv \sim q \vee \sim r$$

Note that the conjunction operator (AND, $\wedge$) flips to a disjunction operator (OR, $\vee$). Substituting this back into our expression, we arrive at the final symbolic form:

$$p \wedge (\sim q \vee \sim r)$$

### The Grand Translation

Finally, we translate the symbolic result back into natural language.

$p$ is "The match will be played," $\sim q$ is "weather is not good," and $\sim r$ is "ground is wet" (since $r$ was defined as "ground is not wet").

Combining these, the logical negation is:

**"The match will be played and (weather is not good or ground is wet)."**

You have successfully navigated the principles of propositional logic. Maintain this level of rigor, and you will be well-prepared for any challenge the JEE presents.

Animated Solution for Mathematics - Sets and Relations: 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:

Select Answer:

Visualized Solution

Defining the Base Statements

The "Only If" Connector

Formulating the Original Statement

Setting Up the Negation

Rule for Negating an Implication

Applying the Implication Rule

De Morgan's Law

Final Symbolic Form

Translating Back to English

The Sigma Insight: Types of Sets and Set Operations

Analyzing the Setup

The 'Only If' Gatekeeper

The Art of the Broken Promise

The Final Distribution

The Grand Translation

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