SECTION 13.1 Counting 911 Counting plays a major role in many diverse areas, such as probability, statistics, and computer science; counting techniques are a part of a branch of mathematics called combinatorics . 1 Find All the Subsets of a Set We begin by reviewing the ways in which two sets can be compared. • If two sets A and B have precisely the same elements, we say that A and B are equal and write = A B. • If each element of a set A is also an element of a set B, we say that A is a subset of B and write ⊆ A B. • If ⊆ A B and ≠ A B, we say that A is a proper subset of B and write ⊂ A B. • If ⊆ A B, every element in set A is also in set B, but B may or may not have additional elements. If ⊂ A B, every element in A is also in B, and B has at least one element not found in A. • Finally, the empty set, ∅, is a subset of every set; that is, ∅ ⊆ A A for any set Finding All the Subsets of a Set Write down all the subsets of the set { } a b c , , . Solution EXAMPLE 1 To organize the work, write down all the subsets with no elements, then those with one element, then those with two elements, and finally those with three elements.This gives all the subsets. Do you see why? 0 Elements 1 Element 2 Elements 3 Elements ∅ { } { } { } a, b, c { } { } { } a, b , b, c , a, c { } a, b, c In Words The notation ( ) n A means “the number of elements in set A. ” 13.1 Counting Now Work the ‘Are You Prepared?’ problems on page 915. • Sets ( Section A.1 , pp. A1 – A3 ) PREPARING FOR THIS SECTION Before getting started, review the following: OBJECTIVES 1 Find All the Subsets of a Set (p. 911) 2 Count the Number of Elements in a Set (p. 911) 3 Solve Counting Problems Using the Multiplication Principle (p. 913) Now Work PROBLEM 9 2 Count the Number of Elements in a Set As you count the number of students in a classroom or the number of likes on a YouTube video, what you are really doing is matching, on a one-to-one basis, each object to be counted with the set of counting numbers, n 1, 2, 3, , , … for some number n. If a set A matched up in this fashion with the set 1, 2, , 25 , { } … you would conclude that there are 25 elements in the set A. The notation ( ) = n A 25 is used to indicate that there are 25 elements in the set A. Because the empty set has no elements, we write ( )∅ = n 0 If the number of elements in a set is a nonnegative integer, the set is finite . Otherwise, it is infinite . We shall concern ourselves only with finite sets.
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