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Order and InequalitiesOne important property of real numbers is that they can be ordered. If a and b are real numbers, a is less than b if b - a is positive. This order is denoted by the inequality a < b. The statement “b is greater than a†is equivalent to saying that a is less than b. When three real numbers a, b, and c are ordered such that a < b and b < c we say that b is between a and c and a < b < c. Geometrically, a < b if and only if a lies to the left of b on the real line (see the figure below).
For example, 1 < 2 because 1 lies to the left of 2 on the real line. The following properties are used in working with inequalities. Similar properties are obtained if < is replaced by ≤ and > is replaced by ≥. (The symbols ≤ and ≥ mean less than or equal to and greater than or equal to, respectively.) Properties of Inequalities Let a, b, c, d, and k be real numbers.
NOTE Note that you reverse the inequality when you multiply by a negative number. For example, if x < 3 then -4x > -12. This also applies to division by a negative number. Thus, if -2x > 4, then x < -2. A set is a collection of elements. Two common sets are the set of real numbers and the set of points on the real line. Many problems in calculus involve subsets of one of these two sets. In such cases it is convenient to use set notation of the form {x: condition on x}, which is read as follows.
For example, you can describe the set of positive real numbers as
Similarly, you can describe the set of nonnegative real numbers as
The union of two sets A and B, denoted by is the set of elements that are members of A or B or both. The intersection of two sets A and B, denoted by is the set of elements that are members of A and B. Two sets are disjoint if they have no elements in common. The most commonly used subsets are intervals on the real line. For example, the open interval
is the set of all real numbers greater than a and less than b, where a and b are the endpoints of the interval. Note that the endpoints are not included in an open interval. Intervals that include their endpoints are closed and are denoted by
The nine basic types of intervals on the real line are shown in the table below. The first four are bounded intervals and the remaining five are unbounded intervals. Unbounded intervals are also classified as open or closed. The intervals (-∞, b) and (a, ∞) are open, the intervals (-∞, b] and [a, ∞) are closed, and the interval (-∞, ∞) is considered to be both open and closed. |
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