- Set inclusion.
- Other operations on sets: exponentiation (power set) and cartesian product.
- Relations.
- Composition of two relations:
Given a relation R between A and B, and a relation S
between B and C, the composition of R and S, denoted by
(R o S), is defined as:
(R o S) = {(x,z) | exists y s.t. (x,y) is in R and (y,z) is in S}

- Inverse: Given a relation R between A and B, the inverse of
R is the relation R
^{-1}between B and A defined byR

^{-1}= {(x,y) | (y,x) is in R} - Equivalence relations.
- Equivalence class:
Given a set A, an equivalence relation R on A,
and an element x of A, the equivalence class of x w.r.t.
R is
[x]

Quotient set: The quotient set of A w.r.t. is_{R}= {y in A | x R y}[A]

_{R}= {[x]_{R}| x is in A } - Partition:
Given a set A, a partition P of A is a set of subsets of A
such that
- for every X, Y in P, X and Y are disjoint (i.e. their intersection is empty)
- the union of al the sets in P is equal to A.

**Theorem (Correspondence between equivalence relations and partitions)**Given an equivalence relation R on A, [A]_{R}is a partition of A. Viceversa, if P is a partition of A, then the relationR = {(x,y) | exists X in P s.t. x, y are in X }

is an equivalence relation on A (and the quotient of A w.r.t. R is exactly P).- Function: a function f : A -> B is a relation between A and B
such that
**(1)**For every x in A, and y_{1}, y_{2}in B, if both (x,y_{1}) and (x,y_{2}) are in f, then y_{1}= y_{2}.

If, in addition to**(1)**, f satisfy the property**(2)**For every x in A, there exists y in B s.t. f(x) = y

**Note:**There is some confusion about terminology: Mathematicians usually call "functions" the relations which satisfy both**(1)**and**(2)**, and call "partial functions" the relations which satisfy only**(1)**(and not**(2)**). We adopt the convention most used in Computer Science, which is to call "functions" (or "partial functions") the relations which satisfy**(1)**(regardeless of whether they satisfy**(2)**or not) and "total functions" the relations which satisfy both**(1)**and**(2)**. In this convention, clearly, the total functions are a subset of the partial functions. -
**Problem:**What are the conditions on f for- f
^{-1}to be a function - f
^{-1}to be a total function

- f