2.2  Mathematical Notation

We indicate that an element a is contained in a set A with the notation a A. We indicate that a set B is a subset of A with the notation B A. If B  A and A  B, the sets A and B are equal, A = B. We denote the union and intersection of two sets A₁ and A₂ as A₁ A₂ and A₁  A₂, respectively.

We use the following notation to indicate familiar sets of numbers:

• : natural numbers {1, 2, 3, …};
• : integers { … , −2, −1, 0, 1, 2, … };
• : real numbers.

In Section 2.5, we introduce the set of complex numbers, which we denote:

• : complex numbers.

We may indicate the elements of a set by listing them between brackets. The set of natural numbers less than 5 can be expressed {1, 2, 3, 4}. Shorthand notion for a set {x₁, x₂, x₃, … , x_n} is simply {x_i}, which is read “the set of x_i“. We may use the notation {x A: property} to indicate the subset of A whose elements satisfy the indicated property. In this manner, the set of natural numbers less than 5 can be expressed (x : x < 5}. If the set A is evident from context, the notation may be simplified as {x: property}.

We denote intervals of real numbers with parentheses or braces, depending upon whether or not end points are included. The interval (2, 3) is the set {x : 2 < x < 3}. The interval [5, 10) is the set {x : 5 ≤ x < 10}.