# What does “$\cong$” sign represent?

I came across this sign when reading some papers. I looked up Wikipedia. It says "The symbol "$\cong$" is often used to indicate isomorphic algebraic structures or congruent geometric figures." So if A $\cong$ B, does this mean A and B are roughly the same but not equal?

Edit: I found this sign in a paper call Identifying Change Points in Linear Regressionns http://goo.gl/dNMONQ, In page 9 there is an equation(equation 3.1) says RSS $\cong$ RSS1+RSS2. RSS is residual sum of square of one regression line, RSS1 and RSS2 is another two residual sum of square of regression lines.

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What is the context in which it was asked? Yes it means congruency relationship, if 2 triangles are congurent, they are exactly equal. –  abstractnature Feb 13 at 7:29
Whats the difference between "≅" and "="? –  user3226059 Feb 13 at 7:35
Do you know what isomorphic means? –  5xum Feb 13 at 7:38
Depends on the context. If one is talking about groups, $D_3$ and $S_3$ have identical "structure", so $D_3 \cong S_3$, but they are not equal, strictly speaking. ($D_3$ is the symmetry group of the triangle, and $S_3$ is the set of permutations on three elements) –  Henry Swanson Feb 13 at 7:39
@5xum it means two things are of the same shape, am I right? –  user3226059 Feb 13 at 7:42

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:

1. 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.

2. 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.

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great response! –  A.E Feb 13 at 8:11

It means that the two structures $A$ and $B$ are isomorphic, but what exactly that means depends entirely on what kind of structures we are talking about.

For instance if $A$ and $B$ are groups, then $A\cong B$ means that $A$ and $B$ are isomorphic as groups. This means that viewed as groups they are essentially identical, they share all the same group properties. Explicitly, if $A$ has group operation $*$ and $B$ has operation $\star$ then $A\cong B$ means that there exists a bijective function (one-to-one and onto) $f:A\rightarrow B$ such that $f(a_1*a_2)=f(a_1)\star f(a_2)$ for all $a_1,a_2\in A$. The map $f$ is called an isomorphism. From the existence of such a map, one can deduce that if $A$ has a certain group property, so does $B$, and vice-versa.

The type of isomorphism that $\cong$ indicates is completely context dependent, it could be an isomorphism of rings, groups, sets, Modules, Algebras etc.

Conversely, $A=B$ usually is used to indicate that $A$ and $B$ are the same, which is stronger than isomorphic. If $A=B$ then one has no need to construct an isomorphism $f$ to deduce the properties of $A$ from the properties of $B$ because they are identical.

For example if we are talking just about set theory We can look at $A=\{1,2,3\}$ and $B=\{2,1,3\}$. Then $A=B$. If instead we have $A=\{1,2,3\}$ and $B=\{2,4,5\}$, then $A\neq B$. but $A\cong B$ as sets because one can define a map $f:A\rightarrow B, 1\mapsto 4, 2\mapsto 2, 3\mapsto 5$, this function is both one-to-one and onto and so since sets possess no additional structures, we can say that $A$ is isomorphic to $B$ as sets.

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In abstract algebra, $\cong$ indicates that two structures are isomorphic to each other.

In simple terms, if $A$ and $B$ are defined by sets elements and some properties describing interactions between elements, $A$ and $B$ are isomorphic to each other if $A$ can be made to look exactly like $B$ by simply re-labeling the elements.

The function that re-labels the elements is called an isomorphism.

In general, an isomorphism is a structure preserving map between two algebraic structures which admits an inverse (wikipedia).

Your first exposure to the concept of isomorphism should be between "groups" in abstract algebra, whereby a group isomorphism is a function which is bijective (one-to-one and onto) and homomorphic.

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