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Prove or disprove some set relations

If A = \{ 1 \} and B = \{ 1,2 \} prove or disprove the following:

  1. A \subset B.
  2. A \subseteq B.
  3. A \in B.
  4. 1 \in A.
  5. 1 \subseteq A.
  6. 1 \subset B.

  1. True Proof. To show A \subset B we must show that every element in A is also in B. Further (since Apostol seems to distinguish between \subset and \subseteq) we want to show that A \neq B.

    First, since 1 is the only element of A, and 1 \in B, we see that every element of A is indeed contained in B. Hence, A \subseteq B.

    Then, since 2 \in B, but 2 \notin A, we see that B \nsubseteq A. Hence A \neq B, so A is a proper subset of B, or A \subset B. ∎

  2. True
    Proof. This follows immediately from part (a) since A \subset B \implies A \subseteq B. ∎
  3. Not True
    This is not true since A = \{ 1 \} and the set \{ 1 \} is not an element of B (since 1 \neq \{ 1 \}).
  4. True
    Proof. There is really nothing to prove here. By definition, A is the set containing 1, so 1 is in A. ∎
  5. Not true
    This is not true since 1 is not a set, so is not a subset of A.
  6. Not true
    Again, 1 is not a set, so cannot be a subset of B.

One comment

  1. Mathieu St-Amour says:

    Why does (b) true when A does’nt equal B ? Isin’nt that the distinction of ? I thought that either one or the other could’ve be true .

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