# Sets

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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:

• Give a precise verbal definition of the set stating what its elements are (e.g. "All of the people residing in New York City as of January 1, 2011 at 12:00 noon"). Obviously, care needs to be taken to ensure that the definition is sufficiently precise. Is the example definition sufficiently precise?
• Make a list of all of the elements in the set, separated by commas and enclosed in brace brackets ({ and }) (e.g. {Joe, Jane, Bob}). This method is only practical for sets containing only a few elements.
• Make a list of a few of the members of the set, separated by commas and enclosed in brace brackets, followed by ... so that the pattern is obvious (e.g. {1, 2, 3, ...}). This method can sometimes be problematic, as there may be disagreement as to whether a pattern is obvious or not.
• Enclose in brace brackets a statement giving the nature of the elements in the set, and a property or condition that must hold for all of the elements in the set but no others (e.g. {x ∈ N: x < 5}).

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.