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).
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}.
To find the intersection of A and (B symmetric difference C), we first need to find the symmetric difference of B and C.
The symmetric difference of two sets, B and C, denoted as B ⊕ C, is the set of elements that are in either B or C, but not in both.
Given:
ℕ = {1, 2, 3, ...} (universal set)
A = {x: x ≤ 6}
B = {x: 4 ≤ x ≤ 9}
C = {2, 3, 5, 7, 8}
We can calculate A ∩ (B ⊕ C) step by step:
Step 1: Find B ⊕ C
To find the symmetric difference of B and C, we need to find elements that are in B or C, but not in both.
B ⊕ C = (B ∪ C) - (B ∩ C)
= ({4, 5, 6, 7, 8, 9} ∪ {2, 3, 5, 7, 8}) - ({5, 7, 8})
= {2, 3, 4, 5, 6, 9}
Step 2: Find A ∩ (B ⊕ C)
To find the intersection of A and (B ⊕ C), we need to find elements that are in both A and (B ⊕ C).
A ∩ (B ⊕ C) = A ∩ {2, 3, 4, 5, 6, 9}
= {1, 2, 3, 4, 5, 6} ∩ {2, 3, 4, 5, 6, 9}
= {2, 3, 4, 5, 6}
Therefore, 𝐴 ∩ (𝐵 ⊕ 𝐶) = {2, 3, 4, 5, 6}.
To find the intersection of set A with the symmetric difference of sets B and C, follow these steps:
Step 1: Find the symmetric difference of sets B and C.
The symmetric difference of two sets, B and C, denoted as B ⊕ C, is the set that contains elements that are in either B or C, but not in both. In this case, B = {x: 4 ≤ x ≤ 9} and C = {2, 3, 5, 7, 8}. So, to find B ⊕ C, we need to check which elements are in either B or C, but not in both.
B ⊕ C = {x: x ∈ B or x ∈ C, but not both}
B ⊕ C = {x: (x ∈ B and x ∉ C) or (x ∉ B and x ∈ C)}
B ⊕ C = {x: (4 ≤ x ≤ 9 and x ∉ C) or (x ∉ B and x ∈ C)}
B ⊕ C = {x: (4 ≤ x ≤ 9 and x ∉ {2, 3, 5, 7, 8}) or (x ∉ {4, 5, 6, 7, 8, 9} and x ∈ {2, 3, 5, 7, 8})}
After applying the conditions, we find that B ⊕ C = {4, 6, 9}.
Step 2: Find the intersection of set A with the symmetric difference of B and C.
To find A ∩ (B ⊕ C), we need to find the elements that are present in both set A and the symmetric difference of B and C.
A = {x: x ≤ 6}
(A ∩ (B ⊕ C)) = {x: (x ≤ 6) and (x ∈ {4, 6, 9})}
After applying the conditions, we find that A ∩ (B ⊕ C) = {4, 6}.
Therefore, 𝐴 ∩ (𝐵 ⊕ 𝐶) = {4, 6}.