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The question provides three sets:

- \( M = \{x \ | \ x \in \mathbb{I}, -3 < x \leq 2\} \), where \( \mathbb{I} \) stands for integers.
- \( N = \{x \ | \ x \in \mathbb{W}, x < 5\} \), where \( \mathbb{W} \) stands for whole numbers.
- \( O = \{x \ | \ x \in \mathbb{N}, 3 \leq x \leq 7\} \), where \( \mathbb{N} \) stands for natural numbers.

We are asked to prove the following two set equalities:

1. \( (O \cup N) \cup M = O \cup (N \cup M) \)
2. \( (M \cup N) \cup O = M \cup (N \cup O) \)

These proofs are essentially demonstrating the **associative property** of union for sets. Let's break them down:

### (i) Proof of \( (O \cup N) \cup M = O \cup (N \cup M) \)
- The union operation is **associative**, which means that no matter how we group the sets, the result remains the same.
- \( (O \cup N) \cup M \) can be simplified by first finding \( O \cup N \), then taking the union with \( M \).
- \( O \cup (N \cup M) \) similarly involves first finding \( N \cup M \), then taking the union with \( O \).

Since union is associative, we know that:
\[
(O \cup N) \cup M = O \cup (N \cup M)
\]

### (ii) Proof of \( (M \cup N) \cup O = M \cup (N \cup O) \)
- Similar to the first part, this is another application of the associative property of union.
- \( (M \cup N) \cup O \) can be simplified by first taking \( M \cup N \), then the union with \( O \).
- \( M \cup (N \cup O) \) can be simplified by first taking \( N \cup O \), then the union with \( M \).

Again, by the associative property of union:
\[
(M \cup N) \cup O = M \cup (N \cup O)
\]

Thus, both statements are true due to the **associativity** of the union operation.

Would you like more detailed steps, or do you have any specific questions? Here are some follow-up questions you might find helpful:

1. What are the key properties of set operations, such as union and intersection?
2. How would you approach proving associativity in other set operations like intersection?
3. Can we extend these properties to more than three sets?
4. How does the commutative property of union help in set operations?
5. What are some practical applications of these set properties?

**Tip**: Remember, associative properties mean that the grouping of elements does not affect the outcome, which simplifies many proofs in set theory.

Let M = {x | x ∈ I and -3 < x ≤ 2}, N = {x | x ∈ W and x < 5}, O = {x | x ∈ N, and 3 ≤ x ≤ 7}. Then, prove that: (i) (O ∪ N) ∪ M = O ∪ (N ∪ M) and (ii) (M ∪ N) ∪ O = M ∪ (N ∪ O)

In this problem, students are tasked with proving the associative property of union for three sets M, N, and O. The sets are defined based on integers, whole numbers, and natural numbers. This problem helps solidify understanding of key set theory concepts such as union and associative laws.

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