“ ������������������������������������������������������������������������������������������������������������������������������������������������������������ ������������������������������������������������������������������������������������������������������������������������������������������������������������ ����������������” ���������������������������������������������������������������� SECTION 1.1 Sets and the Real Numbers In Chapter 1 we will discuss the concepts of sets, the set of real numbers, operations with real numbers, numerical and algebraic expressions. How would you classify the following numbers? 5 hr: the approximate time the average American watches TV each day -273°C: the temperature of absolute zero 6 1 3 ft: the height of the tallest U.S. President, Abraham Lincoln 16π ft2: the area of a circle with radius 4 ft The numbers 5, -273, 6 1 3 , and 16π are examples of real numbers. We encounter real numbers on a daily basis. In this section, we will discuss the set of real numbers in detail. Sets A set is a collection of objects. Each object in a set is called a member or an element. A set is written in braces, { }, and is usually denoted with a capital letter. Sets can either be finite or infinite. A finite set has a specific number of elements. An example of a finite set is A = 51, 2, 3, 4, 56. An infinite set has infinitely many elements. An example of an infinite set is B = 51, 3, 5, 7, 9, . . .6. The three dots at the end are called an ellipsis and indicate that the set continues in the same manner. In preceding sets A and B, the elements of the sets are listed explicitly. When we use this method to represent a set, we are using the roster method. Another method to represent elements in a set is called set-builder notation. In this method, we state the conditions the elements must satisfy; set-builder notation describes the members of a set but does not list them. Set-builder notation is written in the form: {x u condition x must satisfy} This is read as “the set of x, such that, ______.” In the blank, we insert the condition that x must satisfy. The letter x is a variable and represents some unknown number. We will discuss variables in the next section. An example is C = {x u x is a positive odd number}. The condition stated tells us that x can be 1, 3, 5, 7, 9, . . . . This is the same as the previous set B. A set that contains no elements is called the empty set and is denoted by 5 6 or . As we talk about sets, there is some terminology and notation that we should learn. To illustrate the notation, let A = 53, 4, 56 and B = 53, 4, 5, 6, 76. Symbol Meaning Example Verbal Statement ∈ Element 4 ∈ A “4 is an element of A.” or “4 is a member of A.” ∉ Is not an element of 6 ∉ A “6 is not an element of A.” or “6 is not a member of A.” ⊂ Subset A ⊂ B “Set A is contained in set B.” This means that every element in A is also contained in B. ▶ OBJECTIVES As a result of completing this section, you will be able to 1. Understand terminology related to sets and perform operations on sets. 2. Classify a number as a natural number, whole number, integer, rational number, or irrational number. 3. Graph real numbers on a real number line. 4. Find the opposite of a real number. 5. Find the absolute value of a real number. 6. Troubleshoot common errors. Objective 1 ▶ Understand terminology related to sets and perform operations on sets. 2
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