Let's break down the problem step by step.

We are given:

• $A = \{ x : 2x + 1, x \in \mathbb{N}, x < 5 \}$.

We first determine the set $A$:

Since $x \in \mathbb{N}$ (the set of natural numbers) and $x < 5$, we have $x \in \{1, 2, 3, 4\}$.

Now, let's calculate $2x + 1$ for each value of $x$:

• For $x = 1$, $2x + 1 = 2(1) + 1 = 3$.
• For $x = 2$, $2x + 1 = 2(2) + 1 = 5$.
• For $x = 3$, $2x + 1 = 2(3) + 1 = 7$.
• For $x = 4$, $2x + 1 = 2(4) + 1 = 9$.

Thus, $A = \{ 3, 5, 7, 9 \}$.

Next, let’s understand the expression $(A \cup B) - (A \cap B) = (A - B) \cup (B - A)$, where $B$ is some other set.

We can apply set operations based on this identity:

• $A \cup B$ is the union of sets $A$ and $B$,
• $A \cap B$ is the intersection of sets $A$ and $B$,
• $A - B$ is the set difference, i.e., elements in $A$ but not in $B$,
• $B - A$ is the set difference, i.e., elements in $B$ but not in $A$.

The formula $(A \cup B) - (A \cap B) = (A - B) \cup (B - A)$ is known as the distributive property of set operations, and it holds true for any sets $A$ and $B$.

If you'd like to proceed with a specific set $B$, I can demonstrate this equality further with an example. Would you like to define set $B$? Or do you have any questions on the theory behind this equality?

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Here are some related questions to explore:

1. What is the meaning of the set difference $A - B$?
2. How do we compute the union and intersection of two sets?
3. Can you provide an example where this distributive property is not true?
4. How would the formula change if we added a universal set $U$ to the equation?
5. What is the difference between a set and a multiset?

Tip: Always remember to use Venn diagrams when visualizing set operations. It’s a powerful tool for understanding relations like union, intersection, and difference!