Question

If \(A = \{1,2,3,4,5\}\), then which of the following statements is not true?

• A. \(\exists x\in A\), such that \(x+4\leq 9\)
• B. \(\exists x\in A\), such that \(12-x = 7\)
• C. \(\forall x\in A,x+3\geq 4\)
• D. \(\exists x\in A\), such that \(x+2 < 3\)

Hint:

Test each statement against the five elements of $A$.

Answer 1

The Correct Option is D

Step 1: Understand the quantifiers
\(\exists\) means "there is at least one \(x\) in \(A\)" and \(\forall\) means "every \(x\) in \(A\)". We test each statement on \(x=1,2,3,4,5\).

Step 2: Check (A)
\(x+4\le9\) holds for \(x=5\) (and all others). So (A) is true.

Step 3: Check (B)
\(12-x=7\) gives \(x=5\), which is in \(A\). So (B) is true.

Step 4: Check (C)
The smallest value of \(x+3\) is \(1+3=4\ge4\), so it holds for all \(x\). So (C) is true.

Step 5: Check (D)
\(x+2<3\) means \(x<1\). No element of \(A\) is below \(1\). So (D) is false, and it is the statement that is not true.

Final Answer:
Only statement (D) fails, option (D). \[ \boxed{\text{(D)}} \]