Question

What is a relation? Is it the same thing as a graph? Definition says a relation is the mapping between a set of input values and a set of output values.

Answer 1

a relation is just a set of ordered pairs

Answer 2

which defines the relation
so if you see \((a,b)\) as am ordered pair, that means \(a\) relates to \(b\)

Answer 3

*an ordered pair

Answer 4

Hmmm... The definition of a relation is apparently a bit controversial in the math world.

One definition says that a relation is exactly its graph: that it is simply a set of ordered tuplets (of ordered pairs for a binary relation, that is, a relation between exactly two sets or a set and itself).

The other definition says that a relation is slightly more than just its own graph. This definition says that a (binary) relation is an ordered triple (we can of course extend this to all relations, if we add more places in the ordered tuplet), \((X,Y,G)\), where X is the domain of the relation, Y is the codomain of the relation, and G is the graph of the relation (\(G=\{(x_i,y_i):x\in X\wedge y\in Y\}\) ).

The real difference between these two definitions is demonstrated in the following example:
\[G_1=\{(x\in \mathbb{R},y\in \{0,1\}):x=1\}\]\[G_2=\{(x\in \mathbb{Z},y\in \{0,1\}):x=1\}\]By the first definition, these two graphs correspond to the same relation. (The graphs are equivalent.) By the second definition, these two graphs correspond to two different relations.