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Matching set union with set intersection

This seems like it should be a very simple concept to understand, but I just don’t seem to be able to wrap my head around it.
If I have set $X = \{a,b,c,d,e,f,g,h\}$
and I have set $Y = \{a,b,c\}$
how do I determine if the union of the two sets is equal to the intersection of the two sets?
That is, if I am correct in saying that $Z = \{a,b,c,d,e,f,g,h\}$, then $Z$ is equal to $X \cup Y$ and $X \cap Y$ is $Y$, which is not the case.
Why is this?

A:

For sets $X$ and $Y$, $X \cup Y$ is the same set as $X$ or $Y$ (depending on what you look for in a collection)
$X \cap Y$ is the same set as $X$ or $Y$, so $X$ and $Y$ may or may not be equal.
The thing to be careful of is that for a set $X$, we have that $X \cup X = X$ and $X \cap X = X$. For this reason, to check if $X$ and $Y$ are equal, we should instead check if $X$ is a subset of $Y$ and if $Y$ is a subset of $X$:

$X$ is a subset of $Y$ if $X$ is a subset of every element of $Y$: $X \subseteq Y$.
$Y$ is a subset of $X$ if $X$ is a subset of every element of $Y$: $X \subseteq Y$.

In your case, $X = \{a,b,c,d,e,f,g,h\}$ and $Y = \{a,b,c\}$ and $Y \subseteq X$, but

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