# Help: If A$\subseteq$B proof C-B$\subseteq$C-A

• ozarga
In summary, the problem is to prove that when A is a subset of B, then C-B is a subset of C-A. The solution involves using the definitions of set complement and contraposition, and ultimately results in showing that C-B is indeed a subset of 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

## 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

ozarga said:

## 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

## 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

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 .

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$\in$Bc$\rightarrow$x$\in$Ac // 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?

## What does "A$\subseteq$B" mean in the statement "Help: If A$\subseteq$B proof C-B$\subseteq$C-A"?

In set theory, A$\subseteq$B means that every element in set A is also an element of set B. In other words, A is a subset of B.

## What does "C-B" represent in the statement "Help: If A$\subseteq$B proof C-B$\subseteq$C-A"?

C-B represents the set of elements that are in set C but not in set B. This is also known as the relative complement of B in C.

## What is the meaning of "C-A" in the statement "Help: If A$\subseteq$B proof C-B$\subseteq$C-A"?

C-A represents the set of elements that are in set C but not in set A. This is also known as the relative complement of A in C.

## What is the purpose of proving C-B$\subseteq$C-A in this statement?

This statement is asking to prove that the set of elements that are in C but not in B is a subset of the set of elements that are in C but not in A. In other words, it is asking to show that the relative complement of B in C is contained within the relative complement of A in C.

## What are some real-life examples that can help understand this statement better?

One example could be a Venn diagram where set A represents all the students in a school and set B represents all the students who play a sport. If we take the relative complement of B in A, it would represent the students who do not play a sport. On the other hand, if we take the relative complement of A in B, it would represent the students who play a sport but do not attend this particular school. Thus, proving C-B$\subseteq$C-A would show that all the students who do not play a sport are also not attending the school.

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