Set Language

{5, 7}

Step 1: Find A ∪ B first. A∪B = {1,2,3,4,6,8}. Step 2: (A∪B)' = U − (A∪B) = elements in U but NOT in A∪B. Step 3: U = {1,2,3,4,5,6,7,8}. Remove {1,2,3,4,6,8}. Step 4: Remaining elements = {5, 7}. So (A∪B)' = {5, 7}. Common mistake: Finding A' ∪ B' instead of (A∪B)'. Note: By De Morgan's law, (A∪B)' = A'∩B' = {1,3,5,7}∩{5,6,7,8} = {5,7}. Same answer. ✓

{1, 2}

Step 1: A − B means elements in A that are NOT in B. Step 2: A = {1,2,3,4,5} and B = {3,4,5,6,7}. Step 3: Elements of A: check each — 1∉B ✓, 2∉B ✓, 3∈B ✗, 4∈B ✗, 5∈B ✗. Step 4: A − B = {1, 2}. Common mistake: {1,2,6,7} is A△B (symmetric difference), not A−B. {6,7} would be B−A.

3

Step 1: Use the formula n(A∪B) = n(A)+n(B)−n(A∩B). Step 2: Substitute: 18 = 12 + 9 − n(A∩B). Step 3: 18 = 21 − n(A∩B). Step 4: n(A∩B) = 21 − 18 = 3. Common mistake: Adding 12+9 to get 21 and thinking that is n(A∪B), or not rearranging the formula correctly.

{a, b, e, f}

Step 1: A△B = (A−B) ∪ (B−A). Find each part. Step 2: A−B = elements in A not in B = {a, b} (since c,d are in B). Step 3: B−A = elements in B not in A = {e, f} (since c,d are in A). Step 4: A△B = {a,b} ∪ {e,f} = {a,b,e,f}. Common mistake: {c,d} is the intersection, not the symmetric difference. {a,b,c,d,e,f} is the union.