Question

The negation of the statement \(\exists x \in A\) such that \(x+5>8\) is

• A. \(\forall x \in A,\; x+5 \ge 8\)
• B. \(\forall x \in A,\; x+5 \le 8\)
• C. \(\forall x \in A,\; x+5>8\)
• D. \(\exists x \in A\) such that \(x+5<8\)

Hint:

Negation always switches \(\exists\) with \(\forall\) and reverses the inequality sign.

Answer 1

The Correct Option is B

Step 1: Understand the given statement.
The given statement is: \[ \exists x \in A \text{ such that } x+5>8 \] This means “there exists at least one element \(x\) in set \(A\) for which \(x+5\) is greater than 8”. 

Step 2: Apply rules of negation. 
The negation rules are: \[ \neg(\exists x\, P(x)) = \forall x\, \neg P(x) \] \[ \neg(x+5>8) = x+5 \le 8 \] 
Step 3: Write the negated statement. 
\[ \forall x \in A,\; x+5 \le 8 \] 
Step 4: Conclusion. 
Hence, the correct negation is \[ \forall x \in A,\; x+5 \le 8 \]