Given sets A and B, prove that A is a subset of B (Apostol)

[privyet]

Homework Statement

Continuing with my Apostol efforts. From Section I 2.5:

These exercises go over some of the absolute basics of sets. In 3. I'm given A = {1}, B = {1,2} and asked to decide whether some statements are true or false, proving the ones that are true. Seeing which ones are true is no problem but I have no idea what the proper way to 'prove' them might be. For example, How do I prove that A is a subset of B

The Attempt at a Solution

Assume A is a subset of B, then for any x in A, x is in B...no idea

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Another question (6.) gives us A = {1,2} and C = {{1}, {1,2}} and it asks if A is an element of C if so to prove it. I see that it is but once again have no idea how to prove it.

 

[Slats18]

Definition of a subset:

A set U is a subset of X, U ⊂ X, if
∀z (z ∈ U → z ∈ X).

Are you able to manipulate that to answer your question?

 

[HallsofIvy]

The standard way to prove "A is a subset of B" is to prove "if x is in A then x is in B". If you are given that A= {1} and B= {1, 2}, then:

if x is in A, x= 1. 1 is in B. Therefore A is a subset of B.

To prove A is NOT a subset of B is easier- you just need a counter example- find one member of A that is not in B. If A= {1} and B= {{1}, {1, 2}} A is NOT a subset of B because x= 1 is in A but not in B (whose member are sets of numbers, not numbers.

To show that A is a member of B, just note that A= {1} so B= {A, {1, 2}}.

 

[privyet]

Thank you both. I'm afraid that I wasn't able to get to far using just the definition. Not really having any experience with proofs it is difficult to know what is sufficient, it seems that sometimes something ostensibly very simple or obvious has quite a complicated proof, so it's good to see the 'right' way to do it.