1. proving subsets

Let A, B, and C be sets. Show that A U B is a subset of (A U B U C).

I am not sure how exactly to go about doing this. The idea I had was to show that they were equivalent by using definitions of complements, unions...etc. but they are not equal sets.

I am assuming the method of proof used for these problems is using definitions of intersection, complement, unions...etc. And set identities.

Thank you

2. Begin as follows:

Let $\displaystyle x\in A \cup B$. Then $\displaystyle x\in A$ or $\displaystyle x\in B$.

Can you take it from here?

3. I still don't see the light at the end of the tunnel with just that step. I kind of poked around doing something like with (A U B) that but didn't see how it would show its a subset of (A U B U C). Maybe the next step or two steps will cause a light bulb over the head moment.

4. DrSteve's suggestion is the way to go, but you need to have the intuitive understanding of the situation. Suppose there are three groups of people. One night, the first two groups had a meeting, and the next night, all three groups has a meeting. Can you convince another person that everybody who was at the first meeting was also at the second one?

5. Let me add one more comment.

In general to prove that one set is a subset of another set, you start by taking an arbitrary element of the first set, and then you argue that this element is in the second set.

In the example you have given, this is almost obvious. If you start with an element that is either in A or in B, then obviously it's in either A or B or C.

6. I understand the idea. But not sure how it is supposed to be set up. I am not sure how to explain what is troubling me. I think I learn better through reverse learning. Seeing how it is done then ask myself why it was done that way and just connect the reasoning to everything. Then from doing that I gain an understanding. I am guessing my word problem reading comprehension is very very poor so I do better seeing it visually done. lots of show examples with brief sentences explaining the visual steps.

I think seeing the next 1 or 2 steps will help me get the idea (maybe). I am just predicting. I might have an idea already but totally unsure if its even correct. It might be the case that I see the answer and then go "Oh, thats what it meant for me to do?". Then in theory I should be on the same neural wavelength to be able to do the rest of the sub problems.

7. Let $\displaystyle x\in A \cup B$. Then $\displaystyle x\in A$ or $\displaystyle x\in B$. So $\displaystyle x\in A$ or $\displaystyle x\in B$ or $\displaystyle x\in C$. Thus $\displaystyle x\in A \cup B\cup C$.

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a u b subset aubuc

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