# Which of the following are true and which are false? Let A = {0,1,2,3,4} and let B = P(A) (the power set of A) Confirm my answers.

(a) 1∈A (c) {1}∈A (e) {1}⊆A (g) A∈B

(b) 1∈B (d) {1}∈B (f) {1}⊆B (h) B⊆B

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(a) is true because 1 is an element (member) of A.

(b) is false because the elements of B are precisely the subsets of A, but 1 is an element, not a subset, of A.

(c) is false because {1} is not an element (member) of A; instead, {1} is a subset of A.

(d) is true because the elements of B are precisely the subsets of A, and {1} is a subset of A.

(e) is true because {1} is a subset of (i.e. is contained in) A.

(f) is false because 1 is an element of {1}, but as we saw in b), 1 is not an element of B; so {1} is not a subset of B.

(g) is true because the elements of B are precisely the subsets of A, and clearly A is a subset of itself.

(h) is true since B is clearly a subset of itself.

• Everything you said was correct. – MJD Aug 26 '14 at 23:31

Your answers are great! As a remark, note that (e) and (d) are equivalent statements. In general: $$S \subseteq A \iff S \in P(A)$$