1. **State the problem:** We are given sets: - $U = \{\text{all prime numbers}\}$ - $p = \{x : 8(x + 1) \geq 2 (x + 10)\}$ - $D = \{x : 7 < x < 31\}$ where $x$ is a prime number. We need to find: a) $(p \cup D)'$ b) $p'$ c) $(p \cap D)'$ 2. **Find set $p$:** Solve inequality: $$8(x + 1) \geq 2 (x + 10)$$ $$8x + 8 \geq 2x + 20$$ $$8x - 2x \geq 20 - 8$$ $$6x \geq 12$$ $$x \geq 2$$ Since $x$ is prime, $p = \{x : x \geq 2, x \text{ prime}\} = \{2,3,5,7,11,13,17,19,23,29,31, ...\}$ 3. **Find set $D$:** Given $7 < x < 31$ and $x$ prime, primes between 7 and 31 are: $$D = \{11,13,17,19,23,29\}$$ 4. **Find $p \cup D$: union of $p$ and $D$:** Since $p$ includes all primes $\geq 2$, and $D \subset p$, $$p \cup D = p$$ 5. **Find complement of $p \cup D$ with respect to $U$: $(p \cup D)'$:** Since $p \cup D = p$ and $U$ is all primes, $$(p \cup D)' = p' = U \setminus p = \{ \text{prime numbers not in } p \}$$ But $p$ includes all primes $\geq 2$, and the smallest prime is 2, so no prime is excluded. Hence, $$(p \cup D)' = \emptyset$$ 6. **Find $p'$ complement:** As above, $$p' = \emptyset$$ 7. **Find $p \cap D$: intersection:** Since $D \subset p$, $$p \cap D = D = \{11,13,17,19,23,29\}$$ 8. **Find $(p \cap D)'$: complement wrt $U$:** $$(p \cap D)' = U \setminus D = \{\text{primes not in } D\} = \{2,3,5,7,31,37,41, ...\}$$ **Final answers:** a) $(p \cup D)' = \emptyset$ b) $p' = \emptyset$ c) $(p \cap D)' = \{2,3,5,7,31,37,41, ...\}$ (all primes except those in $D$)