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\subseteqB, then it happens that C-B\subseteqC-A

Homework Equations

The Attempt at a Solution

This is how I have been trying to solve it:
1. A\subseteqB // Hyp
2. x\inA\rightarrowx\inB //Element wise proof [1]
3. x\inA \wedge x\inB // which rule allow me to do this? [2]
4. ( x\inA \wedge x\inB) \vee x\inC // Addition [3]
5. (x\inA\veex\inC)\wedge(x\inB\veex\inC) // Distrivutive law [4]
6. x\inA\veex\inC // \wedge OUT [5]
7. x\inB\veex\inC // \wedge OUT [5]
8. y\inC // 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\notinB // Modus Ponendo Tollens [7,8]
10. y\inC\wedgey\notinB // \wedge IN [8,9]
11. y\notinA // Modus Ponendo Tollens [6,8]
12. y\inC\wedgey\notinA // \wedge IN [8,11]
13. y\inC\wedgey\notinB \rightarrow y\inC\wedgey\notinA // 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\subseteqB, then it happens that C-B\subseteqC-A

Homework Equations

The Attempt at a Solution

This is how I have been trying to solve it:
1. A\subseteqB // Hyp
2. x\inA\rightarrowx\inB //Element wise proof [1]
3. x\inA \wedge x\inB // which rule allow me to do this? [2]
4. ( x\inA \wedge x\inB) \vee x\inC // Addition [3]
5. (x\inA\veex\inC)\wedge(x\inB\veex\inC) // Distrivutive law [4]
6. x\inA\veex\inC // \wedge OUT [5]
7. x\inB\veex\inC // \wedge OUT [5]
8. y\inC // 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\notinB // Modus Ponendo Tollens [7,8]
10. y\inC\wedgey\notinB // \wedge IN [8,9]
11. y\notinA // Modus Ponendo Tollens [6,8]
12. y\inC\wedgey\notinA // \wedge IN [8,11]
13. y\inC\wedgey\notinB \rightarrow y\inC\wedgey\notinA // 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\subseteqB // hyp
2. x\inA \rightarrow x\inB // for element proof
3. x\notinB \rightarrow x\notinA // Contraposition
4. x\inB^c\rightarrowx\inA^c // Definition of Set Complement in 3
5. x\inC-B\rightarrowx\inC-A // Definition of Complement in 4
6. C-B\subseteqC-A

Is it right?