Question

3. Suppose ℕ = {1,2,3, … } is the universal set and 𝐴 = {𝑥: 𝑥 ≤ 6}, 𝐵 = {𝑥: 4 ≤ 𝑥 ≤ 9} and
𝐶 = {2,3,5,7,8}. Then, find 𝐴 ∩ (𝐵 ⊕ 𝐶).

Answer 1

To find 𝐴 ∩ (𝐵 ⊕ 𝐶), we need to first find the symmetric difference, 𝐵 ⊕ 𝐶.

The symmetric difference, 𝐵 ⊕ 𝐶, is defined as the set of elements that are in 𝐵 or 𝐶, but not both.

From the given sets, 𝐵 = {4,5,6,7,8,9} and 𝐶 = {2,3,5,7,8}.

To find 𝐵 ⊕ 𝐶, we need to find the elements that are in 𝐵 or 𝐶 but not in both:

𝐵 ⊕ 𝐶 = {4,5,6,7,8,9} ⊕ {2,3,5,7,8}
= {4,6,9} ⊕ {2,3}

Now we can find the intersection of this symmetric difference with set 𝐴.

𝐴 = {𝑥: 𝑥 ≤ 6} = {1,2,3,4,5,6}

𝐴 ∩ (𝐵 ⊕ 𝐶) = {1,2,3,4,5,6} ∩ {4,6,9}
= {4,6}

Therefore, 𝐴 ∩ (𝐵 ⊕ 𝐶) = {4,6}.