# Discrete M: Show that if A ⊆ B and C ⊆ D, then A X C ⊆ B X D

1. Oct 18, 2015

### leo255

1. The problem statement, all variables and given/known data

Sorry that I wasn't able to fit everything in the title. I got 2/3 on this on my quiz, and am wondering what I did wrong, or could have done better. Thanks in advance.

Show that if A ⊆ B and C ⊆ D, then A X C ⊆ B X D

2. Relevant equations

3. The attempt at a solution

For a given value x, if A ⊆ B, then x ∈ A and x ∈ B.
For a given value x, if C ⊆ D, then x i∈ C and x ∈ D.

Cartesian product of (A, C) means that all ordered pairs, (a, c) are included.
Cartesian product of (B, D) means that all ordered pairs, (b, d) are included.

A X C ⊆ B X D

2. Oct 18, 2015

### sxal96

Do you know why your instructor docked you points on your quiz?

Here's an alternative route to getting started that's based on what you already have:
For some x ∈ A ⊆ B, then x ∈ A and x ∈ B. Similarly, for some y ∈ C ⊆ D, then y ∈ C and y ∈ D.

Cartesian products don't necessarily comprise of (x,x); we have to assume that there are two arbitrary elements of the two products, hence why I used (x,y).. We assume the statement above is true based off of what you are given to believe is true, which is that A ⊆ B and C ⊆ D. What conclusion can you draw from what we just stated in the italics?

3. Oct 18, 2015

### HallsofIvy

Staff Emeritus
You really haven't proven anything! You start by stating some definitions (always a good start) then simply assert the conclusion.

To prove "$X\subset Y$" start with "if $p\in X$" and use the definitions of X and Y to conclude "therefore $p \in Y$". Here $X= A\times C$. Now, if $p\in A\times C$, what can you say about p?

4. Oct 18, 2015

### LCKurtz

That isn't the definition of A ⊆ B. Never mind that the statement isn't even true. You might start by looking up the correct definition of A ⊆ B.

5. Oct 18, 2015

### leo255

Yes, you are correct - Math is not my strongest area, and I did not state that correctly. A being a subset means that A is a part of B (i.e. it is contained in B). Also, it is a proper subset if it is not equal to B.

6. Oct 19, 2015

### LCKurtz

While that is an informal understanding, it is not the definition, and you need to use the correct definition to prove your proposition. The statement that A is a subset of B means if $a \in A$ then $a \in B$. So for your problem, you need to show, step by step, using what you are given, that if $p \in A\times C$ then $p \in B\times D$.

7. Oct 19, 2015

### HallsofIvy

Staff Emeritus
If $x\in A$ then $x\in B$

If $x\in c$ then $x\in D$