Answer:
Explanation:
o prove the first associative law of set theory, we need to show that for any sets A, B, and C:
A ∪ (B ∪ C) = (A ∪ B) ∪ C
To do this, we need to show that any element that is in the left-hand side of the equation is also in the right-hand side, and vice versa.
First, let's consider an arbitrary element x.
If x ∈ A ∪ (B ∪ C), then x must be in A, or in B, or in C (or in two or more of these sets).
If x ∈ A, then x ∈ A ∪ B, and so x ∈ (A ∪ B) ∪ C.
If x ∈ B, then x ∈ B ∪ C, and so x ∈ A ∪ (B ∪ C), which means that x ∈ (A ∪ B) ∪ C.
If x ∈ C, then x ∈ B ∪ C, and so x ∈ A ∪ (B ∪ C), which means that x ∈ (A ∪ B) ∪ C.
Therefore, we have shown that if x ∈ A ∪ (B ∪ C), then x ∈ (A ∪ B) ∪ C.
Next, let's consider an arbitrary element y.
If y ∈ (A ∪ B) ∪ C, then y must be in A, or in B, or in C (or in two or more of these sets).
If y ∈ A, then y ∈ A ∪ (B ∪ C), and so y ∈ (A ∪ B) ∪ C.
If y ∈ B, then y ∈ A ∪ B, and so y ∈ A ∪ (B ∪ C), which means that y ∈ (A ∪ B) ∪ C.
If y ∈ C, then y ∈ (A ∪ B) ∪ C.
Therefore, we have shown that if y ∈ (A ∪ B) ∪ C, then y ∈ A ∪ (B ∪ C).
Since we have shown that any element that is in the left-hand side of the equation is also in the right-hand side, and vice versa, we can conclude that:
A ∪ (B ∪ C) = (A ∪ B) ∪ C
This proves the first associative law of set theory.