Need help understanding a fundamental concept.
Let's consider only the set $V =\{1,2,3,4\}$. Suppose I have two graphs $G,G'$: $$G = \{\{1,3\},\{2,4\}\}\qquad G' = \{\{1,4\},\{2,3\}\} $$ It is obvious these graphs are isomorphic. One possible isomorphism $$1\mapsto 2,\quad 2\mapsto 1,\quad 3\mapsto 3,\quad 4\mapsto 4 $$
Inspired by This a comment under the main post reads as follows: "..As far as I know, two graphs are equal iff they are isomorphic..."
But I just expressed my two graphs as sets and these sets are NOT equal hence these graphs, although isomorphic, are also not equal. I don't know what to believe.
Furthermore, if we say "distinct graphs of $n$ vertices" do we in fact say graphs that are NOT equal?
EDIT: Fixed hyperlink, didn't even notice :(
Usually, graph theory does not care about mere differences in how you label the nodes, so from a graph-theoretic point of view, your $G$ and $G'$ are "the same" graph. This is what the concept of isomorphic graphs expresses.
On the other hand, $G$ and $G'$ are represented by different set-theoretic objects, so it can be confusing to call them "equal", even though the difference between them is not one we care about.
It is nevertheless somewhat common in many areas of mathematics to use the word "equal" about structures that are merely isomorphic, if one has faith that the reader/listener will understand (or guess) that what one really means is "isomorphic". Sometimes that faith is misplaced. Use this language at your own risk.
The two graphs are, however, clearly different as subgraphs of $K_4$, though.