1.02 Sets of Numbers

* A collection of objects is called a set. * The members of any set are called elements. * Elements can be numbers, objects, people, or anything. * When the elements are written as a list, they appear between two braces like these { }.

* Some sets a have a specific number of elements. * Some may have an unlimited number of elements. * Others may have no elements at all. Each of these sets has a special name.

Types of Sets

Sets that have a specific number of elements are called finite sets. For example, assume that you own a set of dishes. The elements in that set could be the soup bow, salad plate, dinner plate, cup, and saucer. This finite set has exactly five elements.

Finite Set: {soup bowl, salad plate, dinner plate, cup saucer}

Suppose you were asked to list every element of the set of integers. Seem impossible? That’s because it is. The set of integers has an endless numer of elements. A set that had an infinite number of elements is called infinite set. An ellipses (…) is used to show that the elements of the set continue on in the same pattern.

Infinite Set: {…, -2, -2, 0, 1, 2…}

Finally, suppose you were asked to write the set of positive integers less than zero. Seem impossible again? That’s right; only this time it’s not because the set has an infinite number of elements. This set has no elements; there are no positive integers less than zero. A set that has no elements is called the empty, or null set. The null set is expressed using empty braces {} or the symbol ∅.

Null Set: {} or ∅

Sets Notation:

Vebal M is the set of natural numbers greater than 5

Roster M = {6,7,8,9,10,…}
Set-builder M = {x| x ∈ N and x > 5}

* A rational number is any number that can be expressed as a fraction. * This includes integers, terminating decimals, and repeating decimals. Examples: 2/3 = 2/3 4 = 4/1 -0.3 = -3/10
0.2(repeated) = 2/9

* Numbers that cannot be expressed as a fraction are called irrational numbers. * Irrational numbers have decimal representations that never repeat and never end.

Examples: π √2 √3
Note: not all square roots are irrational. For example, √9 = 3, which is a rational number.

* Real numbers may fit more than one classification. * For example, the real number 2 is a rational number, an integer, a whole number, and a counting number. * The real numbers are the union of the set of rational numbers with the set of irrational numbers.

B = {-12, -11, -6, 9, 10}

A