2.2 Subsets 53 Definition: Proper Subset Set A is a proper subset of set B, symbolized by A B, , if and only if all the elements of set A are elements of set B and set ≠ A B set (that is, set B must contain at least one element not in set A ). Proper Subsets Solution a) All the elements of set A are contained in set B; therefore A B. # b) The elements 5 and 7 are in set A but not in set B; therefore A BÜ ( A is not a subset of B ). In this example, however, all the elements of set B are contained in set A; therefore, B A. # c) There are fruits, such as bananas, that are in set A that are not in set B, so A B. Ü d) All the elements of set A are contained in set B, so A B. # Note also that B A. # In fact, set = A B set . 7 Now try Exercise 5 MATHEMATICS TODAY The Ladder of Life Amelandfoto/Shutterstock In biology, the science of classifying all living things is called taxonomy. More than 2000 years ago, Aristotle formalized animal classification with his “ladder of life”: higher animals, lower animals, higher plants, lower plants. Today, living organisms are classified into six kingdoms (or sets) called animalia, plantae, archaea, eubacteria, fungi, and protista. Even more specific groupings of living things are made according to shared characteristics. The groupings, from most general to most specific, are kingdom, phylum, class, order, family, genus, and species. For example, a zebra, Equus burchelli , is a member of the genus Equus , as is the horse, Equus caballus . Both the zebra and the horse are members of the universal set called the kingdom of animals and the same family, Equidae; they are members of different species ( E. burchelli and E. caballus ), however. Why This Is Important Scientists use sets to classify and categorize animals, plants, and all forms of life. These sets make it easier to understand the behaviors and characteristics of living organisms. Consider the sets = A {red, blue, yellow} and B {red, orange, yellow, green, = blue, violet}. Set A is a subset of set B, A B, # because every element of set A is also an element of set B. Set A is also a proper subset of set B, A B, , because set A and set B are not equal. Now consider = C {car, bus, train} and = D {train, car, bus}. Set C is a subset of set D, C D, # because every element of set C is also an element of set D. Set C, however, is not a proper subset of set D, C D÷ because set C and set D are equal sets. Example 2 A Proper Subset? Determine whether set A is a proper subset of set B. a) = A {jazz, pop, hip hop} = B {classical, jazz, pop, rap, hip hop} b) = = A B {Netflix, Paramount, Max, Hulu} {Max, Netflix, Hulu, Paramount} Solution a) All the elements of set A are contained in set B, and sets A and B are not equal; thus, A B. , b) Set = A B set , so A B. ÷ (However, A B. # ) 7 Now try Exercise 9 Every set is a subset of itself, but no set is a proper subset of itself. For all sets A, A A, # but A A. ÷ For example, if = A {1, 2, 3}, then A A# because every element of set A is contained in set A, but A A ÷ because set = A A set . Let = A { } and = B {1, 2, 3, 4}. Is A B? # To show A B, Ü you must determine at least one element of set A that is not an element of set B. Because this cannot be done, A B# must be true. Using the same reasoning, we can show that the empty set is a subset of every set, including itself .
RkJQdWJsaXNoZXIy NjM5ODQ=