Question

Select and write the correct answer of the following multiple choice type questions:
(i) If \( A = \{1, 2, 3, 4, 5\} \), then which of the following is not true?

• A. \(\exists x \in A\) such that \(x + 3 = 8\)
• B. \(\exists x \in A\) such that \(x + 2 < 9\)
• C. \(\forall x \in A, x + 6 \geq 9\)
• D. \(\exists x \in A\) such that \(x + 6 < 10\)

Hint:

Remember that universal quantifiers (\(\forall\)) require the statement to be true for all elements in the set.

Answer 1

The Correct Option is C

Step 1: Check each option.
- Option (i): \(\exists x \in A\) such that \(x + 3 = 8\). This is true for \(x = 5\), since \(5 + 3 = 8\). - Option (ii): \(\exists x \in A\) such that \(x + 2 < 9\). This is true for \(x = 1, 2, 3, 4\), as \(1 + 2 = 3\), \(2 + 2 = 4\), \(3 + 2 = 5\), \(4 + 2 = 6\). - Option (iii): \(\forall x \in A, x + 6 \geq 9\). This is false. For \(x = 1\), \(1 + 6 = 7\), which is less than 9. Hence, this is not true for all \(x\). - Option (iv): \(\exists x \in A\) such that \(x + 6 < 10\). This is true for \(x = 1, 2\), since \(1 + 6 = 7\) and \(2 + 6 = 8\).

Step 2: Conclude.
The statement in option (iii) is not true, as there exists at least one \(x\) (i.e., \(x = 1\)) for which \(x + 6 < 9\).

Final Answer: \[ \boxed{(iii) \; \forall x \in A, \; x + 6 \geq 9} \]