Mathematics deals with objects of very different kinds; from your previous experience,  you are familiar with many of them: Numbers, points, lines, planes, triangles,  circles, angles, equations, functions and many more. Often, objects of a similar nature or with a common property are collected into sets; these may be finite or infinite (For the moment, it is enough if you have an intuitive understanding of finite,  resp. infinite; a more rigorous definition will be given at a later stage). The objects which are collected in a set are called the elements of that set. If an object a is an element of a set M, we write

              a ∈ M

which is read as a (is an) element of M.

If a is not an element of M, then we write

                          a ∉M

which is read as a is not an element of M.

Example 1.1.. If Q is the set of all quadrangles, and A is a parallelogram,  then A ∈ Q. If C is a circle, then C ∉Q.

2. If G is the set of all even numbers, then 16 ∈ G, and 3 ∉G.

3. If L is the set of all solutions of the equation x² = 1, then 1 is an element of L, while 2 is not.

Generally, there are two ways to describe a set:

• By listing its elements between curly brackets and separating them by commas,

e.g.

{0},

{2, 67, 9},

{x,y, z}.

Note that this is convenient (or indeed possible) only for sets with relatively few elements. If there are more elements and one wants to list the elements  “explicitly” sometimes periods are used; for example,

                {0, 1, 2, 3,. ..}, {2, 4, 6,. .. , 20}.

The meaning should be clear from the context. In this descriptive or explicit method, an element may be listed more than once, and the order in which the elements appear is irrelevant. Thus, the following all describe the same set:

                  {1, 2, 3}, {2, 3, 1}, {1, 1, 3, 2, 3}.

• By giving a rule which determines if a given object is in the set or not; this is also called implicit description; for example,

1. {x : x is a natural number}

2. {x : x is a natural number and x > 0}

3. {y : y solves (y + 1) · (y − 3) = 0}

4. {p : p is an even prime number}.

Usually, there is more than one way of describing a set. Thus, we could have written

1. {0, 1, 2,...}

2. {1, 2, 3,...}

3. {−1, 3}

4. {2} or {x : 2x = 4}

The general situation can be described as follows: A set is determined by a defining property P of its elements, written as

                        {x : P(x)}

where P(x) means that x has the property described by P. The letter x serves as a variable for objects; any other letter, or symbol except P, would have done equally well; similarly, P is a variable for properties or, as they are sometimes called,  predicates. To avoid running into logical difficulties we shall always assume that our objects which are described by the predicate P come from a previously well defined set, say M, and sometimes we shall say so explicitly. In general then, we describe sets by

                        {x : x ∈ M and P(x)},

which also can be written as

                       {x ∈ M : P(x)}.

We shall use the following conventions in describing certain sets of numbers:

• N = {0, 1, 2, 3,.. .} is the set of natural numbers.

• N⁺ = {1, 2, 3,. ..} is the set of positive natural numbers.

• Z = {... , −3, −2, −1, 0, 1, 2, 3,.. .} is the set of integers.

• Q = {x : x =a/b }, where a ∈ Z,b ∈ Z, and b ≠0, is the set of rational numbers; observe that each rational number is the ratio of two integers,  whence the name.

• R = {x : x is a real number}.