Statement I: Given two sets A and B, if A ∪ B = A ∩ B, then it must mean that A = B.
Statement II: A ∪ B = {x | x ∈ A or x ∈ B} and A ∩ B = {x | x ∈ A and x ∈ B}.
Statement I is true but Statement II is false.
Statement I is false but Statement II is true.
Both Statement I and Statement II are true.
Both Statement I and Statement II are false.
Answer
If A ∪ B = A ∩ B, then every element of A belongs to A ∪ B = A ∩ B and hence belongs to B. Similarly, every element of B belongs to A. So A = B.
∴ Statement I is true.
By definition, A ∪ B = {x | x ∈ A or x ∈ B} and A ∩ B = {x | x ∈ A and x ∈ B}.
∴ Statement II is true.
Hence, option 3 is the correct option.
Statement I: If A = {x | x is a digit in the number system} and B = {x | x is a single digit multiple of 3}, then A - B = {1, 2, 4, 5, 7, 8}.
Statement II: A - B = {x | x ∈ A or x ∉ B}.
Statement I is true but Statement II is false.
Statement I is false but Statement II is true.
Both Statement I and Statement II are true.
Both Statement I and Statement II are false.
Answer
A = {x | x is a digit in the number system}
A = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
B = {x | x is a single digit multiple of 3}
B = {3, 6, 9}
A - B = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} - {3, 6, 9}
A - B = {0, 1, 2, 4, 5, 7, 8} ≠ {1, 2, 4, 5, 7, 8}
∴ Statement I is false.
Statement II: By definition, A - B = {x | x ∈ A and x ∉ B}, not {x | x ∈ A or x ∉ B}.
∴ Statement II is false.
Hence, option 4 is the correct option.
Statement I: Given any set A, A ∪ φ = A and A ∩ φ = φ.
Statement II: A and φ are disjoint sets.
Statement I is true but Statement II is false.
Statement I is false but Statement II is true.
Both Statement I and Statement II are true.
Both Statement I and Statement II are false.
Answer
As the empty set φ has no elements, A ∪ φ = A and A ∩ φ = φ.
∴ Statement I is true.
As A ∩ φ = φ, sets A and φ have no element in common, so A and φ are disjoint sets.
∴ Statement II is true.
Hence, option 3 is the correct option.