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Help!: If A[itex]\subseteq[/itex]B proof C-B[itex]\subseteq[/itex]C-A

  1. Mar 25, 2012 #1
    1. The problem statement, all variables and given/known data
    Hello, I'm having problems with the following exercise from my homework.

    Proof that when A[itex]\subseteq[/itex]B, then it happens that C-B[itex]\subseteq[/itex]C-A


    2. Relevant equations



    3. The attempt at a solution
    This is how I have been trying to solve it:
    1. A[itex]\subseteq[/itex]B // Hyp
    2. x[itex]\in[/itex]A[itex]\rightarrow[/itex]x[itex]\in[/itex]B //Element wise proof [1]
    3. x[itex]\in[/itex]A [itex]\wedge[/itex] x[itex]\in[/itex]B // which rule allow me to do this? [2]
    4. ( x[itex]\in[/itex]A [itex]\wedge[/itex] x[itex]\in[/itex]B) [itex]\vee[/itex] x[itex]\in[/itex]C // Addition [3]
    5. (x[itex]\in[/itex]A[itex]\vee[/itex]x[itex]\in[/itex]C)[itex]\wedge[/itex](x[itex]\in[/itex]B[itex]\vee[/itex]x[itex]\in[/itex]C) // Distrivutive law [4]
    6. x[itex]\in[/itex]A[itex]\vee[/itex]x[itex]\in[/itex]C // [itex]\wedge[/itex] OUT [5]
    7. x[itex]\in[/itex]B[itex]\vee[/itex]x[itex]\in[/itex]C // [itex]\wedge[/itex] OUT [5]
    8. y[itex]\in[/itex]C // Hyp (This step is the very confusing one. I'm assuming it exists an element of one set I have no information it actually exists)
    9. y[itex]\notin[/itex]B // Modus Ponendo Tollens [7,8]
    10. y[itex]\in[/itex]C[itex]\wedge[/itex]y[itex]\notin[/itex]B // [itex]\wedge[/itex] IN [8,9]
    11. y[itex]\notin[/itex]A // Modus Ponendo Tollens [6,8]
    12. y[itex]\in[/itex]C[itex]\wedge[/itex]y[itex]\notin[/itex]A // [itex]\wedge[/itex] IN [8,11]
    13. y[itex]\in[/itex]C[itex]\wedge[/itex]y[itex]\notin[/itex]B [itex]\rightarrow[/itex] y[itex]\in[/itex]C[itex]\wedge[/itex]y[itex]\notin[/itex]A // CP [10, 12]
    14. C-B [itex]\subseteq[/itex] C-A // Defs of Difference [13] & Element wise proof

    Sorry for my poor english. Thanks in advance for your help.
     
  2. jcsd
  3. Mar 25, 2012 #2

    SammyS

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    Generally, to prove that [itex]C-B\subseteq C-A\,,[/itex] you take an element of set C-B and show that it is an element of set C-A .
     
  4. Mar 25, 2012 #3
    Thank you SammyS.

    I think I figured out:

    1. A[itex]\subseteq[/itex]B // hyp
    2. x[itex]\in[/itex]A [itex]\rightarrow[/itex] x[itex]\in[/itex]B // for element proof
    3. x[itex]\notin[/itex]B [itex]\rightarrow[/itex] x[itex]\notin[/itex]A // Contraposition
    4. x[itex]\in[/itex]Bc[itex]\rightarrow[/itex]x[itex]\in[/itex]Ac // Definition of Set Complement in 3
    5. x[itex]\in[/itex]C-B[itex]\rightarrow[/itex]x[itex]\in[/itex]C-A // Definition of Complement in 4
    6. C-B[itex]\subseteq[/itex]C-A

    Is it right?
     
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