Question

If \(A = \{1,2,3,4,5,6\}\) then which of the following is not true?

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

Hint:

Test each statement against the set; a "for all" statement fails if one element breaks it.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
\(\exists\) needs at least one element that works. \(\forall\) needs every element to work. A is \(\{1,2,3,4,5,6\}\).

Step 2: Check (A):
\(x+3=9\) gives \(x=6\), which is in A. TRUE.

Step 3: Check (B):
\(x+2<9\) is true for \(x=1\). TRUE.

Step 4: Check (C):
\(x+6\geq10\) needs \(x\geq4\). Take \(x=1\): \(7\geq10\) is false. So the "for all" statement is FALSE.

Step 5: Check (D):
\(x+6<10\) is true for \(x=1\) (7 is less than 10). TRUE.

Step 6: Conclusion:
The statement that is not true is (C).

Final Answer:
Statement C is not true. \[ \boxed{\text{(C) }\forall x\in A,\ x+6\geq10} \]