2.6 Infinite Sets 87 Which set is larger, the set of integers or the set of even integers? One might argue that because the set of even integers is a subset of the set of integers, the set of integers must be larger than the set of even integers. Yet both sets are infinite sets, so how can we determine which set is larger? This question puzzled mathematicians for centuries until 1874, when Georg Cantor developed a method of determining the cardinal number of an infinite set. In this section, we will discuss infinite sets and how to determine the number of elements in an infinite set. SECTION 2.6 Infinite Sets LEARNING GOALS Upon completion of this section, you will be able to: 7 Understand infinite sets. 7 Understand countable sets. Why This Is Important The concept of infinity plays an important role in many applications of mathematics and science. For example, calculus relies on the concept of infinity, and calculus allows doctors to determine the amount of blood flow in a human heart. Calculus also is used by businesses to make important decisions that can maximize profits and minimize costs. Infinite Sets In Section 2.1, we state that a finite set is a set in which the number of elements is zero or the number of elements can be expressed as a natural number. Also in Section 2.1, we define a one-to-one correspondence. To determine the number of elements in a finite set, we can place the set in a one-to-one correspondence with a subset of the set of counting numbers. For example, the set A {#, ?, $} = can be placed in one-to-one correspondence with set B {1, 2, 3}, = a subset of the set of counting numbers. A B {#, ?, $} {1, 2, 3} = ↓ ↓ ↓ = Because the cardinal number of set B is 3, the cardinal number of set A is also 3. Any two sets, such as set A and set B, that can be placed in a one-to-one correspondence must have the same number of elements (therefore the same cardinality) and must be equivalent sets. Note that n A( ) and n B( ) both equal 3. German mathematician Georg Cantor (1845–1918), known as the father of set theory, thought about sets that were not bounded. He called an unbounded set an infinite set and provided the following definition. Definition: Infinite Set An infinite set is a set that can be placed in a one-to-one correspondence with a proper subset of itself. In Example 1, we use Cantor’s definition of an infinite set to show that the set of counting numbers is infinite. m Georg Cantor
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