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

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.

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?

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LCKurtz
Homework Helper
Gold Member

## Homework Statement

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

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
Homework Helper

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