Determine which statements are false or true. Justify your answer.

Question

Determine which statements are false or true. Justify your answer.

(a) {0, 1} ∈ {0, {0}, {1}, 1}

(b) {a, b, c} ⊂ {a, {a, b}, c, {a, c}, {{{a, b, c}}}}

(c) ∅ ⊆ {a, b, c}

(d) {1} ∈ {{1}, 0, {{1}}}

Answer 1

Hi Isaac,

First, recall the distinction between ∈ and ⊆.

A ∈ B indicates that A is an element of B, i.e. A appears at least once in the set of objects B.

On the other hand, A ⊆ B indicates that A is a subset of B, i.e. all objects within A are contained within the set B. So, for your examples...

a) Our A in this case is {0,1}, and B is the set {0, {0}, {1}, 1}. A does not appear anywhere in this set, so this statement is false. If B instead was {0, {0}, {1}, 1, {0,1}}, then it would be true.

b) Now, we are asked if the set {a,b,c} is a strict subset of the set B={a, {a, b}, c, {a, c}, {{{a, b, c}}}}. {a,b,c} is clearly contained in B, and there are other elements within B aside from {a,b,c}. Thus the statement is true.

c) This one is a bit more conceptual. We are asked if the empty set ∅ is a subset of {a,b,c}. The answer for this is always true. If we were instead asked if ∅∈{a,b,c}, then this would be false since the empty set is not an element of {a,b,c}.

So, can you use this information from parts a, b, and c to answer d)?