For one of my homework problems, we are told to prove if this statement is correct. I'm not sure how to approach this problem. I was thinking of drawing a venn diagram, but I'm not sure if this is possible. Are we supposed to use set identities?
For any sets X, Y, Z, if X ∪ Z ⊆ Y ∪ Z, then X ⊆ Y.
On
For elementary set theory problems like this, you can play with the definition.
$A \subset B$ if and only if any element $a$ in $A$ is also an element of $B$.
Now, let us see the definition of $X \cup Z$. This is equivalent to say $a \in X \cup Z \equiv a \in X \vee a \in Z$. Next, we also see (by similar way) that $b \in Y \cup Z \equiv b \in Y \vee b \in Z$.
For the next step, take any $x \in X$, then we have either $x \in Z$ or $x \not\in Z$. If $x \not\in Z$, then (left as an exercise for you) it is obvious $x \in Y$ holds true and hence $X \subset Y$.
Now, if $x \in Z$ too, then $x \in X \wedge x \in Z \equiv x \in X \cap Z$. Please proceed on your own now :)
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Suppose that the statement is true.
Then substituting $X=Z$ it implies the statement that $Z\subseteq Y$ for every set $Y$.
This however can only be true if $Z=\varnothing$.
It is easily checked that $X\cup\varnothing\subseteq Y\cup\varnothing$ indeed implies that $X\subseteq Y$.
So 0ur final conclusion is that the statement is true if and only if $Z=\varnothing$.
Hint for disproof:
Consider two sets $X \neq \emptyset$, $Y \neq \emptyset$ with $X\cap Y = \emptyset$. Set $Z = X \cup Y$.