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

  1. Oct 18, 2015 #1
    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. jcsd
  3. Oct 18, 2015 #2
    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?
     
  4. Oct 18, 2015 #3

    HallsofIvy

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

    To prove "[itex]X\subset Y[/itex]" start with "if [itex]p\in X[/itex]" and use the definitions of X and Y to conclude "therefore [itex]p \in Y[/itex]". Here [itex]X= A\times C[/itex]. Now, if [itex]p\in A\times C[/itex], what can you say about p?
     
  5. Oct 18, 2015 #4

    LCKurtz

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    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.
     
  6. Oct 18, 2015 #5
    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.
     
  7. Oct 19, 2015 #6

    LCKurtz

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    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##.
     
  8. Oct 19, 2015 #7

    HallsofIvy

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    If [itex]x\in A[/itex] then [itex]x\in B[/itex]

    If [itex]x\in c[/itex] then [itex]x\in D[/itex]
     
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