For a set A, the power set of A is denoted by 2^A. If A = {{0,1}, {1}}, which of the following options are TRUE?
I. empty_set subset_of 2^A
II. {1} subset_of 2^A
III. {{1}} in 2^A
IV. {{0,1}} subset_of 2^A
A. I and III only
B. II and III only
C. I, II and III only
D. I, II and IV only
GATE 2015 · Discrete Mathematics · Set Theory · medium
Answer: I, II and III only (option C)
Enumerate 2^A: A = {{0,1}, {1}}. Elements of A are: e1 = {0,1}, e2 = {1}. So 2^A = { {}, {e1}, {e2}, {e1,e2} } = { {}, {{0,1}}, {{1}}, {{0,1},{1}} }.
Check Statement I: empty_set subset_of 2^A: The empty set is a subset of every set by definition. So empty_set subset_of 2^A is TRUE.
Check Statement II: {1} subset_of 2^A: In the context of this problem, {1} denotes the set whose single element is the set {1} (i.e., the set e2). So {1} here = {e2} = {{1}}. Is e2 = {1} an element of 2^A? Yes: {{1}} in 2^A. So {{1}} = {e2} subset_of 2^A iff e2 in 2^A, which is true. Hence {1} subset_of 2^A is TRUE (interpreting {1} as {e2}).
Check Statement III: {{1}} in 2^A: {{1}} in 2^A iff {{1}} is a subset of A. Is {{1}} subset_of A? We need every element of {{1}} to be in A. The only element of {{1}} is {1}. Is {1} in A? Yes, {1} is one of the two elements of A. So {{1}} subset_of A, hence {{1}} in 2^A is TRUE.
Check Statement IV: {{0,1}} subset_of 2^A: {{0,1}} subset_of 2^A iff {0,1} in 2^A iff {0,1} subset_of A. Is {0,1} subset_of A? We need 0 in A and 1 in A. But A = {{0,1},{1}}: 0 is not an element of A (only {0,1} and {1} are elements). So {0,1} is NOT a subset of A. Hence {0,1} NOT in 2^A, so {{0,1}} is NOT a subset of 2^A.