Prove set identity

Question

I find it hard to prove this identity:

Given $A,B,C$ if $A\setminus C = B\setminus C$ and $A\cap C = B\cap C$, then $A\subseteq B$.

I started by taking $x \in A$ and then $x \in A\cup \text{any set}$

but I can't find a connection from there to the given info,

can you help please?

Answer 1

Hint: Take any $x\in A$ and consider the alternative:

• either $x$ also lies in $C$,
• or it doesn't,

and use the hypotheses in each case.

Answer 2

Take $a\in A$. There are two possibilities:

• $a\in C$: then $a\in A\cap C=B\cap C\subset B$.
• $a\notin C$: then $a\in A\setminus C\subset B\setminus C\subset B$.

So, in both cases, $a\in B$.

Answer 3

$\forall x \in A$ either $x \in A\setminus C$ or $x \in A\cap C$

$x \in A\setminus C \implies x\in B\setminus C$ and $x\in B$

$x \in A\cap C \implies x\in B\cap C$ and $x\in B$

Answer 4

In fact, $A = B$:

$$A = (A \cap C) \cup (A \setminus C) = (B \cap C) \cup (B \setminus C) = B$$

Answer 5

I find it hard to prove this identity:

Why?

$A$ is the union of $A\cap C$ and $A\smallsetminus C$.   You have been told that each of these sets are equal to sets which are themselves obviously subsets of $B$.   Therefore $A$ is the union of subsets of $B$ and thus itself a subset of $B$.

A more formal proof will use a "proof by cases" structure.

$$\def\fitch#1#2{~~\begin{array}{|l}#1\\\hline#2\end{array}}\fitch{(A\cap C)=(B\cap C)\\ (A\smallsetminus C)=(B\smallsetminus C)}{\fitch{x\in A}{\fitch{x\in C}{x\in A\wedge x\in C\\~ \\ ~\\ ~\\ x\in B}\qquad\fitch{x\notin C}{x\in A\wedge x\notin C\\~\\~\\ ~\\ x\in B}\\x\in B}\\x\in A\to x\in B\\ A\subseteq B}$$

Fill in the missing steps and provide justifications.