Hard Time Understanding If A C B, then A U B = B

[number0]

Hard Time Understanding "If A C B, then A U B = B"

Homework Statement

Understand how "If A \supseteq B, then A U B = B" is possible.

Homework Equations

None.

The Attempt at a Solution

Since A U B = B, it can be separated into two cases. That is,

1) A U B \subseteq B
2) B \subseteq A U B

For case (1), I let x \in A U B. Thus, either x \in A or x \in B. If x \in A, then x \in B. This means that A U B \subseteq B.

For case (2), I let x \in B. This is where I got stuck... I know I am supposed to apply the assumption that A \supseteq B, but I am starting to think that it is impossible.

Here is an attachment of why I think it is impossible. Here is the link to tinypic for those who are too afraid to download attachments: http://tinypic.com/view.php?pic=xszle&s=7.

Can anyone give me a tip on how to approach this problem?

 

[LCKurtz]

Homework Statement

Understand how "If A \supseteq B, then A U B = B" is possible.

Homework Equations

None.

The Attempt at a Solution

Since A U B = B, it can be separated into two cases. That is,

1) A U B \subseteq B
2) B \subseteq A U B

For case (1), I let x \in A U B. Thus, either x \in A or x \in B. If x \in A, then x \in B. This means that A U B \subseteq B.

For case (2), I let x \in B. This is where I got stuck...

But if x ε B then isn't x ε A U B?

 

[dextercioby]

Understand how "If A \supseteq B, then A U B = B" is possible.

What you wrote holds only if A=B. Another scenario would be A\subseteq B \Rightarrow A\cup B = B

which is trivial.

 

[vela]

Understand how "If A \supseteq B, then A U B = B" is possible.

From the other stuff you wrote, I'm assuming this is a typo and you meant if A ⊂ B, then A ∪ B = B.

In your attachment, you interpreted B ⊂ A ∪ B to mean B is a subset of A and a subset of B. This isn't correct. Say C ⊂ A ∪ B. If x∈C, that only means x∈A or x∈B. It's possible for some elements of C to be only in A and others to be only in B so that C is a subset of neither A nor B. So B ⊂ A ∪ B doesn't tell you anything about whether B is a subset of A or a subset of B. It just says all the elements in B are also elements in the union of A and B.