Question

A={a, b, c, d, e, f, g, x, y, z), B={1, 2, c, d, e), and C={d, e, f, g, 2, v}, does the equation A/(B∪C) = (A/B) ∩ (A/C) hold true?

A. True
B. False

Answer 1

Upon evaluating the sets, it was determined that the equation A/(B∪C) = (A/B) ∩ (A/C) does hold true for the given sets A, B, and C as both sides yield the set {a, b, x, y, z}.

Step-by-step explanation:

The question revolves around set theory and specifically whether the equality A/(B∪C) = (A/B) ∩ (A/C) holds true. Here, / denotes the set difference and ∪ denotes the union of sets, while ∩ refers to the intersection of sets. We will examine if the given equation holds for the provided sets A, B, and C.

Firstly, we need to find the union of sets B and C, which we shall denote as D.

• D = B ∪ C = {1, 2, c, d, e, f, g, v}.

Now, we'll find the difference between set A and the union set D.

• A/D = {a, b, c, d, e, f, g, x, y, z} - {1, 2, c, d, e, f, g, v} = {a, b, x, y, z}.

Next, we'll find the differences between set A and each of the sets B and C separately.

• A/B = {a, b, c, d, e, f, g, x, y, z} - {1, 2, c, d, e} = {a, b, f, g, x, y, z},
• A/C = {a, b, c, d, e, f, g, x, y, z} - {d, e, f, g, 2, v} = {a, b, c, x, y, z}.

Now, we compute the intersection of these two difference sets.

• (A/B) ∩ (A/C) = {a, b, f, g, x, y, z} ∩ {a, b, c, x, y, z} = {a, b, x, y, z}.

Finally, we can clearly see that the set A/(B∪C) is equal to the set (A/B) ∩ (A/C). Thus, the original equation A/(B∪C) = (A/B) ∩ (A/C) holds true for the given sets A, B, and C, and the correct answer is A. True.