# Which of these 4 statements about sets A and B are equivalent?

Which of these 4 statements about sets A and B are equivalent?:

(1) A $$\cap$$ C $$\subseteq$$ B $$\cap$$ C for all C

(2) A $$\subseteq$$ B

(3) A $$\cup$$ C $$\subseteq$$ B $$\cup$$ C for all C

(4) A \ B = $$\emptyset$$

(2) and (4) are obviously equivalent

I think (2) and (3) are also equivalent because x $$\in$$ A $$\cup$$ C $$\iff$$ x $$\in$$ A V x $$\in$$ C $$\rightarrow$$ x $$\in$$ A. Similarly for x $$\in$$ B $$\cup$$ C. So A $$\subseteq$$ B.

But I'm not 100% sure about whether (1) is equivalent to any of the rest. I feel like it is because if I rewrite it into (x $$\in$$ A $$\cap$$ C $$\rightarrow$$ x $$\in$$ B $$\cap$$ C) $$\iff$$ (x $$\in$$ A $$\land$$ x $$\in$$ C $$\rightarrow$$ x $$\in$$ B $$\land$$ x $$\in$$ C)

But can I conclude that x $$\in$$ A $$\rightarrow$$ x $$\in$$ B?

• Please use MathJax also for arrows. Here is a tutorial. Apr 2 at 20:37
• For (1) to be true, one of the $C$ to consider is the empty set, and so $A\cap B\subseteq B\cap \emptyset = \emptyset$. Apr 2 at 20:43
• I thought of that too but then how can I prove (1) and (2) are equivalent for all C? Apr 2 at 20:47
• So (1) and (2) are not equivalent. A counterexample $A = B = \{0\}$, (1) is false (e.g. when $C=\emptyset$), while (2) is true. Apr 2 at 20:52
• If $C=\emptyset$ what does 1) imply? And what does 2) imply? Apr 2 at 21:32

You correctly state that $$(2)$$ is equivalent to $$(4)$$.
Let us show that $$(2)$$ is equivalent to $$(3)$$. First, why $$(2)$$ implies $$(3)$$. Take any $$C$$. Let $$x$$ be in $$A\cup C$$. Then, by definition of union, $$x\in A$$ or $$x\in C$$. Since $$A$$ is a subset of $$B$$, we get that $$x\in B$$ or $$x\in C$$. By definition of union, this means that $$x\in B\cup C$$.
Now let us show that $$(3)$$ implies $$(2)$$. Just take $$C=\emptyset$$. Then $$A\cup C \subseteq B\cup C$$ becomes $$A\subseteq B$$.
$$(1)$$ is also equivalent to $$(2)$$. Why does $$(2)$$ imply $$(1)$$? Take any $$C$$. Now let $$x$$ be in $$A\cap C$$. By definition of intersection, it means that $$x \in A$$ and $$x\in C$$. Since $$A$$ is a subset of $$B$$, $$x$$ is in $$B$$ and in $$C$$. Hence, by definition of intersection, $$x$$ is in $$B\cap C$$.
To see that $$(1)$$ implies $$(2)$$ take $$C=A\cup B$$. Then $$A\cap C \subseteq B\cap C$$ turns into $$A\subseteq B$$.