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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?

Thank you in advance for all your help.

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

1 Answer 1

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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 $A\subseteq B$.

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    $\begingroup$ Thank you very much for your help. I really appreciate it. $\endgroup$
    – Thao Mai
    Apr 3 at 0:06

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