# Is this a clear proof?

1. Sep 30, 2004

### anon1980_1@hotmail.c

Suppose A, B, and C are sets.
Prove that if A union B is a subset of C, then A is a subset of C and B is a subset of C.

My proof:
Suppose A, B, and C are sets such that A union B is a subset of C.
Then for all x, if x is in A union B, then x is in C.
Since x is in A union B, this means x is in A or x is in B.
Then if x is in A or x is in B, then x is in C.
Hence, if x is in A, then x is in C, and if x is in B, then x is in C.
Thus, A is a subset of C and B is a subset of C.

Is this ok?

2. Sep 30, 2004

### matt grime

why not just the obvious:

A < AuB < C

hence A<C

other wise you've got to do tedious propositional logic.

i teach a course like this, and i can't understand why you have to prove something so obvious to be honest. it's bollocks isn't it?

it's obvious and as is often the case with obvious things writing down the proof is a pain, but you're at least on the right track, although you could tidy it up:

to show A<B you need:

x in A true implies

(x in A) or (xi in B) is true

so x in AuB is true,

hence x in C is true by the definition of subset,

3. Sep 30, 2004

### anon1980_1@hotmail.c

Can I just assume that A is contained in A union B which is contained in C?
The way I have written my proof is the way we were taught in class.
Does it make sense? Or are there obvious flaws in the logic?

4. Sep 30, 2004

### Gokul43201

Staff Emeritus
No obvious flaws. Most teachers will accept your proof. A small minority might ask you how "Then if x is in A or x is in B, then x is in C" leads to "Hence, if x is in A, then x is in C, and if x is in B, then x is in C"

5. Sep 30, 2004

### eku_girl83

The next to last line of your proof is a little vague. I'm at a loss for how to make it clearer though.

6. Sep 30, 2004

### anon1980_1@hotmail.c

How can I clarify the next to last line? Someone please offer me suggestions?

7. Oct 1, 2004

### matt grime

I gave a proof that A<AuB

you should try to minimize the number of lines.

Here's my full word proof:

We must show that A<C. Let x be in A

(x in A) => (x in A)or(x in B) => (x in AuB) => (x in C)

and we are done.

if you want more words then write 'which implies that'