# Summary

**Ordered pair:**A pair of elements grouped together in a particular order.**Cartesian product:**A Ã— B of two sets A and B is given by A Ã— B = {(*a*,*b*):*a*âˆˆ*b*âˆˆ

In particular**R**Ã—**R**= {(*x*,*y*):*x*,*y*âˆˆ**R**} and**R**Ã—**R**Ã—**R**= (*x*,*y*,*z*):*x*,*y*,*z*âˆˆ**R**}- If (
*a*,*b*) = (*x*,*y*), then*a*=*x*and*b*=*y.* - If
*n*(A) =*p*and*n*(B) =*q*, then*n*(A Ã— B) =*pq*. - A Ã— Ï† = f
- In general, A Ã— B â‰ B Ã— A.
**Relation:**A relation R from a set A to a set B is a subset of the Cartesian product A Ã— B obtained by describing a relationship between the first element*x*and the second element*y*of the ordered pairs in A Ã— B.- The
**image**of an element*x*under a relation R is given by*y*, where (*x*,*y*) âˆˆ R. - The
**domain** - The
**range**of the relation R is the set of all second elements of the ordered pairs in a relation R. **Function:**A function*f*from a set A to a set B is a specific type of relation for which every element*x*of set A has one and only one image*y*in set B. We write*f*: Aâ†’ B, where*f*(*x*) =*y*.- A is the domain and B is the codomain of
*f*. - The range of the function is the set of images.
- A real function has the set of real numbers or one of its subsets both as its domain and as its range.
**Algebra of functions:**For functions*f*: X â†’**R**and*g*: X â†’**R**, we have

(*f*+*g*) (*x*) =*f*(*x*) +*g*(*x*),*x*âˆˆ(

*f*-*g*) (*x*) =*f*(*x*) -*g*(*x*),*x*âˆˆX.(

*f*.*g*) (*x*) =*f*(*x*) .*g*(*x*),*x*âˆˆ(

*kf*) (*x*) =*k f*(*x*) ),*x*âˆˆ