Let $p, q, r, s$ represent the following propositions. $p: x \in \{8, 9, 10, 11, 12\}$ $q: x\text{ is a composite number}$ $r: x\text{ is a perfect square}$ $s: x\text{ is a prime number}$ The integer $x \ge 2$ which satisfies $eg((p \Rightarrow q) \wedge ( eg r \vee eg s))$ is ________ .

We are given the logical expression: $$eg((p \Rightarrow q) \wedge ( eg r \vee eg s))$$

Applying De Morgan's Law: $$\equiv eg(p \Rightarrow q) \veeeg(eg r \vee eg s)$$

Using the equivalence $eg(p \Rightarrow q) \equiv p \wedge eg q$ and $eg(eg r \vee eg s) \equiv r \wedge s$:$$\equiv (p \wedge eg q) \vee (r \wedge s)$$

Now, evaluate both parts for integers $x \ge 2$:

1. Case 2: $(r \wedge s)$ $r$ means $x$ is a perfect square, and $s$ means $x$ is prime. A prime number cannot be a perfect square for any integer $x \ge 2$, so $r \wedge s$ is always False.

2. **Case 1: $(p \wedge

eg q)$**$p$means$x \in \{8, 9, 10, 11, 12\}$.$ eg q$means$x$is not a composite number. Evaluating each element in the set: -$8$: Composite (False) -$9$: Composite (False) -$10$: Composite (False) -$11$: Prime, hence not composite (True) -$12$: Composite (False)

Thus, the only integer that satisfies the given condition is 11.