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Exercise 2:

Use set-builder notation to describe the sets:

a) {0, 3, 6, 9, 12}

This set contains multiples of 3, starting from 0 up to 12. In set-builder notation: $\{ x \in \mathbb{N} \ | \ x = 3n, \ n \in \mathbb{N} \text{ and } 0 \leq x \leq 12 \}$ where $\mathbb{N}$ is the set of natural numbers.

b) {-3, -2, -1, 0, 1, 2, 3}

This set contains integers between -3 and 3 (inclusive). In set-builder notation: $\{ x \in \mathbb{Z} \ | \ -3 \leq x \leq 3 \}$ where $\mathbb{Z}$ is the set of integers.

Exercise 3:

Determine whether the pairs of sets are equal:

a) {1, 3, 3, 3, 5, 5, 5, 5, 5} and {5, 3, 1}

Yes, these sets are equal because sets ignore duplicates and order. Both simplify to: $\{1, 3, 5\}.$

b) {{1}} and {1, {1}}

No, these sets are not equal. The first set contains only one element, $\{1\}$, while the second set contains two elements: $1$ and $\{1\}$.

c) $\emptyset$ and $\{\emptyset\}$

No, these sets are not equal. The first set is the empty set with no elements, while the second set contains one element, which is the empty set.

Exercise 4:

Determine subsets:

Sets given: $A = \{2, 4, 6\}, \ B = \{2, 6\}, \ C = \{4, 6\}, \ D = \{4, 6, 8\}.$

1. $B \subseteq A$: True (all elements of $B$ are in $A$).
2. $C \subseteq A$: True (all elements of $C$ are in $A$).
3. $D \subseteq A$: False ($8 \notin A$).

Additionally:

• $B \subseteq D$: False ($2 \notin D$).
• $C \subseteq D$: True (all elements of $C$ are in $D$).

Exercise 5:

Determine if 2 or $\{2\}$ is an element of the set:

a) $\{x \in \mathbb{R} : x \text{ is an integer greater than 1}\}$

• $2 \in \text{set}$, because $2$ is an integer greater than 1.
• $\{2\} \notin \text{set}$, because $\{2\}$ is not an integer.

b) $\{x \in \mathbb{R} : x \text{ is the square of an integer}\}$

• $2 \notin \text{set}$, because $2$ is not the square of any integer.
• $\{2\} \notin \text{set}$, because $\{2\}$ is not a number.

c) $\{2, \{2\}\}$

• $2 \in \text{set}$, because $2$ is explicitly listed.
• $\{2\} \in \text{set}$, because $\{2\}$ is explicitly listed.

d) $\{\{2\}, \{\{2\}\}\}$

• $2 \notin \text{set}$, because $2$ is not explicitly listed.
• $\{2\} \in \text{set}$, because $\{2\}$ is explicitly listed.

e) $\{2, \{2\}, \{2, \{2\}\}\}$

• $2 \in \text{set}$, because $2$ is explicitly listed.
• $\{2\} \in \text{set}$, because $\{2\}$ is explicitly listed.

f) $\{\{\{2\}\}\}$

• $2 \notin \text{set}$, because $2$ is not explicitly listed.
• $\{2\} \notin \text{set}$, because $\{2\}$ is not explicitly listed.

Do you need further clarifications or additional explanations?

Related Questions:

1. How do duplicates and order affect equality in sets?
2. Can you explain the difference between $\emptyset$ and $\{\emptyset\}$ in more detail?
3. Why is $\{1\}$ different from $1$ in a set?
4. What are the conditions for a set to be a subset of another?
5. How does set-builder notation vary for different types of numbers?

Tip:

Remember, sets do not care about order or repetition, but they do distinguish between elements and subsets!