Question

If A = {3, 5, 7, 9, 11, 12}, determine the truth value of each of the following.

i) $\exists x \in A$ such that $x - 8 = 1$

ii) $\forall x \in A, x^2 + x$ is an even number

iii) $\exists x \in A$ such that $x^2 < 0$

iv) $\forall x \in A, x$ is an even number

v) $\exists x \in A$ such that $3x + 8 > 40$

vi) $\forall x \in A, 2x + 9 > 14$

Answer 1

(i) True (ii) True (iii) False (iv) False (v) True (vi) True

Answer 2

Let $A = \{3, 5, 7, 9, 11, 12\}$.

1. (i) $\exists x \in A$ such that $x - 8 = 1$.

   • Solve: $x = 9$. Since $9 \in A$, it is True.

2. (ii) $\forall x \in A, \, x^2 + x$ is even.

   • For any integer $x$:
     • If $x$ is odd: odd$^2$ is odd, and odd + odd = even.
     • If $x$ is even: even$^2$ is even, and even + even = even.
   • Since every $x \in A$ (odd or even) yields an even number, it is True.

3. (iii) $\exists x \in A$ such that $x^2 < 0$.

   • Squares of real numbers are never negative. Hence, False.

4. (iv) $\forall x \in A, \, x$ is even.

   • Most elements are odd. So, False.

5. (v) $\exists x \in A$ such that $3x + 8 > 40$.

   • Solve: $3x + 8 > 40$ ⟹ $3x > 32$ ⟹ $x > \frac{32}{3} \approx 10.67$.
   • Elements $11$ and $12$ satisfy this, so True.

6. (vi) $\forall x \in A, \, 2x + 9 > 14$.

   • Solve: $2x + 9 > 14$ ⟹ $2x > 5$ ⟹ $x > 2.5$.
   • Every element in $A$ (all $>2.5$) satisfies this, so True.