16 CHAPTER R Review of Basic Concepts Sets are often written in set-builder notation, which uses a variable, such as x, to describe the elements of the set. The following set-builder notation represents the set 53, 4, 5, 66 and is read “the set of all elements x such that x is a natural number between 2 and 7.” The numbers 2 and 7 are not between 2 and 7. 5x x is a natural number between 2 and 76 = 53, 4, 5, 66 Set-builder notation The set of all elements x such that x is a natural number between 2 and 7 EXAMPLE 1 Using Set Notation andTerminology Identify each set as finite or infinite. Then determine whether 10 is an element of the set. (a) 57, 8, 9, c , 146 (b) E1, 1 4 , 1 16 , 1 64 , cF (c) 5x x is a fraction between 1 and 26 (d) 5x x is a natural number between 9 and 116 SOLUTION (a) The set is finite, because the process of counting its elements 7, 8, 9, 10, 11, 12, 13, and 14 comes to an end. The number 10 belongs to the set. 10∈57, 8, 9, c , 146 (b) The set is infinite, because the ellipsis points indicate that the pattern continues indefinitely. In this case, 10∉E1, 1 4 , 1 16 , 1 64 , cF. (c) Between any two distinct natural numbers there are infinitely many fractions, so this set is infinite. The number 10 is not an element. (d) There is only one natural number between 9 and 11, namely 10. So the set is finite, and 10 is an element. S Now Try Exercises 11, 13, 15, and 17. EXAMPLE 2 Listing the Elements of a Set Use set notation, and list all the elements of each set. (a) 5x x is a natural number less than 56 (b) 5x x is a natural number greater than 7 and less than 146 SOLUTION (a) The natural numbers less than 5 form the set 51, 2, 3, 46. (b) This is the set 58, 9, 10, 11, 12, 136. S Now Try Exercise 25. When we are discussing a particular situation or problem, the universal set (whether expressed or implied) contains all the elements included in the discussion. The letter U is used to represent the universal set. The empty set, or null set, is the set containing no elements. We write the empty set using the special symbol ∅. CAUTION 5∅6 is not the empty set. It is the set containing the symbol ∅.
RkJQdWJsaXNoZXIy NjM5ODQ=