This solution covers multiple math problems involving sets, including listing elements, set-builder notation, and identifying equivalent sets.

here are the solutions to the math problems.

Question (e) from the first image:
Which is the set of prime numbers less than 10?
The prime numbers less than 10 are 2, 3, 5, and 7. (1 is not a prime number).
The correct option is ii. $\{2, 3, 5, 7\}$.

4. A is the set of natural numbers between 0 and 10 exclusive.

• a) List the elements of the set A. Natural numbers start from 1. "Between 0 and 10 exclusive" means numbers greater than 0 and less than 10. $$A = \boxed{\{1, 2, 3, 4, 5, 6, 7, 8, 9\}}$$
• b) Use $\in$ or $\notin$ if 2, 3, 0, 9, 11 belong or do not belong to the set A.
  • $2 \in A$
  • $3 \in A$
  • $0 \notin A$
  • $9 \in A$
  • $11 \notin A$
• c) Write the set in set-builder notation. $$A = \boxed{\{x \mid x \in \mathbb{N}, 0 < x < 10\}}$$

5. a) List the members of the following sets.

• A = {odd numbers from 0 to 10} $$A = \boxed{\{1, 3, 5, 7, 9\}}$$
• B = {prime numbers between 10 and 24} $$B = \boxed{\{11, 13, 17, 19, 23\}}$$
• C = {multiples of 7 between 6 and 50} $$C = \boxed{\{7, 14, 21, 28, 35, 42, 49\}}$$

b) Which sets are equivalent among A, B and C? Equivalent sets have the same number of elements.

• Number of elements in A, $|A| = 5$.
• Number of elements in B, $|B| = 5$.
• Number of elements in C, $|C| = 7$. Sets A and B have the same number of elements. The equivalent sets are $\boxed{A \text{ and } B}$.

6. Given that $P = \{0, 1, 3, 5\}$, $Q = \{5, 3, 1\}$, $R = \{0\}$, $S = \{1\}$, $T = \phi$ and $S = \{3, 1, 5\}$.
Assumption: The second definition of S, $S = \{3, 1, 5\}$, overwrites the first definition $S = \{1\}$.
So, the sets are: $P = \{0, 1, 3, 5\}$, $Q = \{5, 3, 1\}$, $R = \{0\}$, $S = \{3, 1, 5\}$, $T = \phi$.

• a) Write down the number of elements in each set.
  • $|P| = \boxed{4}$
  • $|Q| = \boxed{3}$
  • $|R| = \boxed{1}$
  • $|S| = \boxed{3}$
  • $|T| = \boxed{0}$
• b) Which of the sets are equal? Equal sets contain exactly the same elements. $Q = \{5, 3, 1\}$ and $S = \{3, 1, 5\}$ contain the same elements. The equal sets are $\boxed{Q \text{ and } S}$.
• c) Which of the sets have equal number of elements but the sets are not equal? From part (a), $|Q|=3$ and $|S|=3$. However, Q and S are equal sets. There are no other pairs of sets with an equal number of elements that are not themselves equal. $\boxed{\text{None}}$

7.

• a) If $A = \{a, b, c, d\}$ and $B = \{a, b, c, d, e\}$, all elements of A are also the elements of B, but again $A \neq B$. Why? $A \neq B$ because set B contains an element 'e' that is not present in set A. For two sets to be equal, they must contain exactly the same elements.
• b) If $P = \{g, o, d\}$ and $Q = \{c, u, p\}$, why are P and Q not equal sets? But P is equivalent to Q. Why? P and Q are not equal sets because they do not contain the same elements (e.g., 'g' is in P but not in Q). They are equivalent sets because they have the same number of elements: $|P|=3$ and $|Q|=3$.

8. $A = \{1, 2, 3, 4, 5\}$ is a given set.
Write the following subsets of A (i) by listing (ii) by set-builder notation.

• a) B = Set of elements of A that are prime numbers
  • (i) $B = \boxed{\{2, 3, 5\}}$
  • (ii) $B = \boxed{\{x \mid x \in A \text{ and } x \text{ is a prime number}\}}$
• b) C = Set of elements of A that are even numbers
  • (i) $C = \boxed{\{2, 4\}}$
  • (ii) $C = \boxed{\{x \mid x \in A \text{ and } x \text{ is an even number}\}}$
• c) D = Set of elements of A that are odd numbers
  • (i) $D = \boxed{\{1, 3, 5\}}$
  • (ii) $D = \boxed{\{x \mid x \in A \text{ and } x \text{ is an odd number}\}}$
• d) E = Set of elements of A that are even prime numbers
  • (i) $E = \boxed{\{2\}}$
  • (ii) $E = \boxed{\{x \mid x \in A \text{ and } x \text{ is an even prime number}\}}$
• e) F = Set of elements of A that are between 1 and 4
  • (i) $F = \boxed{\{2, 3\}}$
  • (ii) $F = \boxed{{x \mid x \in A and 1 <