[Math Lair] Sets

Math Lair Home > Topics > Sets

A set is any well-defined collection of objects. Objects in such a collection are called elements or members of that set.

How to Define a Set

There are four ways to define a set:

When specifying a property characterizing the set, the property must be such that it is possible to determine whether or not any object whatsoever has that property or not. If this is the case, then the set is well-defined.

Here are a few sets that are used frequently:
N The set of all natural numbers
Z The set of all integers
Q The set of all rational numbers
R The set of all real numbers
C The set of all complex numbers

If f is a mapping from a set X into a set Y,

  1. If an element x of X is mapped by f to an element y of Y, then y is said to be the image of x.
  2. If every element from Y is the image of some element of X, then the mapping is said to be from X onto Y.
  3. If no element of Y is the image of more than one element of X, then the mapping f is said to be one-to-one.
  4. If every element of X has an image in Y, then the mapping is said to be defined everywhere in X.
  5. If f is onto and defined everywhere in X, then f is called a one-to-one correspondence.

If there exists a one-to-one correspondence between two sets, then we say that the two sets have the same cardinality. If S and T have the same cardinality, then we say that S is equivalent to T, or S ∼ T.

Properties of ∼

  1. Reflexivity: The set S is equivalent to itself (S∼S).
  2. Symmetry: If S is equivalent to T, T is equivalent to S (If S∼T, then T∼S).
  3. Transitivity: If S is equivalent to T and T is equivalent to R, then S is equivalent to R (If S∼T and T∼R, then S∼R).

For more information on sets, see set theory.