Is there a symbol for "homeomorphic to"? I looked on Wikipedia, but it doesn't seem to mention one? Also, for isomorphism, is the symbol a squiggly line over an equals sign? What is the symbol with a squiggly line over just one horizontal line? Thanks.
Notation – Notations Involving Squiggly Lines Over Horizontal Lines
notation
Related Solutions
What you've written, using the horizontal line, is the expression:
$[4 + 2(8 - 3)]/(2-1)$, which is not the original expression. Using the "slash", or horizontal line, we have:
$$4 + \color{blue}{\bf 2(8-3)\div 2} - 1 = 4+\color{blue}{\bf 2(8-3)/2}-1 = 4 + \color{blue}{\bf \frac{2(8-3)}{2}} - 1.$$
In each case in the equality above, the scope of the division by $2$ is restricted to the dividend $2(8 - 3)$.
Very generally, you could think of $\cong$ as meaning "two things are 'essentially the same' but are not identically one", whereas $=$ means they are identically one. This is better understood and explained by looking at the cases where it's used:
In geometry, $\cong$ means congruence of figures, which means the figures have the same shape and size. (In advanced geometry, it means one is the image of the other under a mapping known as an "isometry", which provides a formal definition of what "same shape and size" means) Two congruent triangles look exactly the same, but they are not the same triangle, as then there would be but one triangle, not two. (One could also say they are formed from different points in the space.) Colloquially speaking, they are "copies" or "clones" of each other. A copy of something is not literally identically one and the same as the other, even though they are alike in every respect.
In abstract algebra, $\cong$ means isomorphism, which says the two objects are structurally the same. Intuitively, if we have, say, a pair of groups, and they are isomorphic, the "patterns" formed by the operations are the same, even if the elements making up the groups' base sets are not the same. (That is, if you could draw a table (not physically possible for infinite groups) of the group operations, they would have the same "pattern", just expressed with possibly different symbols) Formally, it means there is a bijective map between the two which respects the operations (in the sense given in the other posts here). Any abstract-algebraic properties of one hold for the other (that is, any properties that do not depend specifically on the particular characteristics of the elements of the base sets as objects in and of themselves). From the point of view of abstract algebra, it is this structure that is what matters, and the precise composition of the base sets does not, so from that point of view we would say they are "essentially the same", but they need not be identically the same, for the composition of the base sets may differ. If the base sets are the same, then we really do have only one mathematical object, and they are equal, $=$.
There is probably some philosophical points here -- the distinction certainly does have a philosophical flavor to it, namely regarding whether or not things which are indiscernible (i.e. $\cong$) are identical (i.e. $=$). In mathematics, it is useful to make a distinction between the two concepts: indiscernible (i.e. congruent) triangles may occupy different parts of space, for example, and isomorphic groups may need to be distinguished if, say, we are dealing with them as part of a larger problem or situation in which we need to deal with more than just their properties from an abstract-algebraic point of view, i.e. where the composition of their base sets is relevant to the problem as well. There might also be cases where there are multiple types of structure defined on the same base set, and two objects may be isomorphic with respect to one kind of structure, but not another.
Best Answer
I use $\cong$ for isomorphism in a category, which includes both homeomorphism and isomorphism of groups, etc. I have seen $\simeq$ used to mean homotopy equivalence, but I don't know how standard this is.