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

[ozarga]

Homework Statement

Hello, I'm having problems with the following exercise from my homework.

Proof that when A$\subseteq$B, then it happens that C-B$\subseteq$C-A

Homework Equations

The Attempt at a Solution

This is how I have been trying to solve it:
1. A$\subseteq$B // Hyp
2. x$\in$A$\rightarrow$x$\in$B //Element wise proof [1]
3. x$\in$A $\wedge$ x$\in$B // which rule allow me to do this? [2]
4. ( x$\in$A $\wedge$ x$\in$B) $\vee$ x$\in$C // Addition [3]
5. (x$\in$A$\vee$x$\in$C)$\wedge$(x$\in$B$\vee$x$\in$C) // Distrivutive law [4]
6. x$\in$A$\vee$x$\in$C // $\wedge$ OUT [5]
7. x$\in$B$\vee$x$\in$C // $\wedge$ OUT [5]
8. y$\in$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$\notin$B // Modus Ponendo Tollens [7,8]
10. y$\in$C$\wedge$y$\notin$B // $\wedge$ IN [8,9]
11. y$\notin$A // Modus Ponendo Tollens [6,8]
12. y$\in$C$\wedge$y$\notin$A // $\wedge$ IN [8,11]
13. y$\in$C$\wedge$y$\notin$B $\rightarrow$ y$\in$C$\wedge$y$\notin$A // CP [10, 12]
14. C-B $\subseteq$ C-A // Defs of Difference [13] & Element wise proof

Sorry for my poor english. Thanks in advance for your help.

 

[SammyS]

Homework Statement

Hello, I'm having problems with the following exercise from my homework.

Proof that when A$\subseteq$B, then it happens that C-B$\subseteq$C-A

Homework Equations

The Attempt at a Solution

This is how I have been trying to solve it:
1. A$\subseteq$B // Hyp
2. x$\in$A$\rightarrow$x$\in$B //Element wise proof [1]
3. x$\in$A $\wedge$ x$\in$B // which rule allow me to do this? [2]
4. ( x$\in$A $\wedge$ x$\in$B) $\vee$ x$\in$C // Addition [3]
5. (x$\in$A$\vee$x$\in$C)$\wedge$(x$\in$B$\vee$x$\in$C) // Distrivutive law [4]
6. x$\in$A$\vee$x$\in$C // $\wedge$ OUT [5]
7. x$\in$B$\vee$x$\in$C // $\wedge$ OUT [5]
8. y$\in$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$\notin$B // Modus Ponendo Tollens [7,8]
10. y$\in$C$\wedge$y$\notin$B // $\wedge$ IN [8,9]
11. y$\notin$A // Modus Ponendo Tollens [6,8]
12. y$\in$C$\wedge$y$\notin$A // $\wedge$ IN [8,11]
13. y$\in$C$\wedge$y$\notin$B $\rightarrow$ y$\in$C$\wedge$y$\notin$A // CP [10, 12]
14. C-B $\subseteq$ C-A // Defs of Difference [13] & Element wise proof

Sorry for my poor english. Thanks in advance for your help.

Generally, to prove that $C-B\subseteq C-A\,,$ you take an element of set C-B and show that it is an element of set C-A .

 

[ozarga]

Thank you SammyS.

I think I figured out:

1. A$\subseteq$B // hyp
2. x$\in$A $\rightarrow$ x$\in$B // for element proof
3. x$\notin$B $\rightarrow$ x$\notin$A // Contraposition
4. x$\in$B^c$\rightarrow$x$\in$A^c // Definition of Set Complement in 3
5. x$\in$C-B$\rightarrow$x$\in$C-A // Definition of Complement in 4
6. C-B$\subseteq$C-A

Is it right?