For any two statements $p$ and $q$ , the negation of the expression $p \lor (\sim p \land q)$ is:

Question

If $ R $ is the smallest equivalence relation on the set $ \{1, 2, 3, 4\} $ such that $ \{(1,2), (1,3)\} \subseteq R $, then the number of elements in $ R $ is ______.

Answer 1

To find the negation of the expression $p \lor (\sim p \land q)$ , we can apply logical negation rules. The original expression can be interpreted in lay terms as follows: "Either p is true, or both p is false and q is true."

First, identify the main components of the expression: it is a disjunction ( $∨$ ) between $p$ and $(∼p∧q)$ .
To negate the expression, use De Morgan's Laws, which tell us that the negation of a disjunction ( $A∨B$ ) is the conjunction of the negations ( $∼A∧∼B$ ). For the given expression, this results in: $∼p∧∼(∼p∧q)$ .
Now, apply negation to the second part $∼(∼p∧q)$ using De Morgan's Laws again. It becomes $p∨∼q$ .
Thus, the expression $∼p∧∼(∼p∧q)$ simplifies to $∼p∧(p∨∼q)$ .
Further simplifying, we apply distribution: $(∼p∧p)∨(∼p∧∼q)$ . The first part $∼p∧p$ is a contradiction and is always false, leaving us with $∼p∧∼q$ .

Hence, the negation of the expression $p∨(∼p∧q)$ is $∼p∧∼q$ , which matches the given correct answer.

Answer 2

To find the negation of the expression $p \lor (\sim p \land q)$ , we can apply logical negation rules. The original expression can be interpreted in lay terms as follows: "Either p is true, or both p is false and q is true."

First, identify the main components of the expression: it is a disjunction ( $∨$ ) between $p$ and $(∼p∧q)$ .
To negate the expression, use De Morgan's Laws, which tell us that the negation of a disjunction ( $A∨B$ ) is the conjunction of the negations ( $∼A∧∼B$ ). For the given expression, this results in: $∼p∧∼(∼p∧q)$ .
Now, apply negation to the second part $∼(∼p∧q)$ using De Morgan's Laws again. It becomes $p∨∼q$ .
Thus, the expression $∼p∧∼(∼p∧q)$ simplifies to $∼p∧(p∨∼q)$ .
Further simplifying, we apply distribution: $(∼p∧p)∨(∼p∧∼q)$ . The first part $∼p∧p$ is a contradiction and is always false, leaving us with $∼p∧∼q$ .

Hence, the negation of the expression $p∨(∼p∧q)$ is $∼p∧∼q$ , which matches the given correct answer.

For any two statements $p$ and $q$ , the negation of the expression $p \lor (\sim p \land q)$ is:

The Correct Option is D

Solution and Explanation

Top Questions on sets

Questions Asked in JEE Main exam

For any two statements p and q, the negation of the expression p∨(∼p∧q) is:

The Correct Option is D

Solution and Explanation

Top Questions on sets

Questions Asked in JEE Main exam

For any two statements $p$ and $q$ , the negation of the expression $p \lor (\sim p \land q)$ is: