Sets Chapter 2 1

Sets: A Problem Solving Tool 2.1

Sets A set is a well-defined collection of objects, called elements or members of the set. Let A = {1, 2, 3, 4}, which means that “A is the set containing the elements 1, 2, 3, and 4.” 4  A “4 is in A” or “4 is an element of the set A” 6  A “6 is not an element of A” 3

Let X = {a, b, x, y}. Fill in the blank with or to make each statement correct. a___X x___X A___X 4

Sets of Numbers Commonly used in Mathematics N = {x | x is a natural number} = {1, 2, 3, …} W = {x | x is a whole number} = {0,1, 2, 3, …} I = {x | x is an integer} = {…, 2, 1, 0,1, 2, 3, …} Q = {x | x is a rational number} = {x | x, is of the form a/b where a, b are integers and b  0} Q = {x | x is a rational number} = {x | x has a terminating or repeating decimal pattern} S = {x | x is a irrational number} = {x | x is not rational} R = {x | x is a real number} = {x | x is the union of the rational and irrational numbers} 5

Describing Sets The Empty Set 1. Verbal or written description Roster Method or Listing Set-builder notation The Empty Set The symbol { } or Ø represents the empty, or null set. NOTE: {Ø} is not the null set. It is the set containing the null set. 6

The set of counting numbers less than 8. Describe the following sets using the roster method and set builder notation. The set of counting numbers less than 8. The set of counting numbers between 3 and 8. The set of counting numbers between 6 and 7. The set of counting numbers greater than 3. a. {1, 2, 3, 4, 5, 6, 7} b. {4, 5, 6, 7} c. { } or φ d. {4, 5, 6,…} 7

Equality of Sets Two sets A and B are equal, A = B, if they have the same elements not necessarily listed in the same order. State whether the sets A and B are equal. A={2n + 1|n is a counting number} B={2n – 1|n is a counting number} A = {1,1,2,2,3} , B = {2,1,3} ANSWER 1. A ≠ B , not equal 2. A = B , equal 8

Subsets Proper Subsets The Universal Set The set A is a subset of B, A ⊆ B, if every element of A is also an element of B. Proper Subsets A set A is said to be a proper subset of B, A ⊂ B, if A ⊆ B but A  B. The Universal Set The universal set, U, is the set of all elements under discussion. 9

List all the subsets and indicate which are the proper subsets of the given set. U = {a, b} U = {1, 2, 3} ANSWER Ø, {a}, {b}, and {a, b}. The first three are proper subsets. Ø {1}, {2}, {3} {1, 2}, {1, 3}, {2, 3} {1, 2, 3} The first seven are proper subsets. Note: by definition Ø is a subset of any set. 10

Number of Subsets of a Set Number of Proper Subsets of a Set A set of n elements has 2n subsets. Refer to the problems on the previous slide: 22 = 4 and 23 = 8. Number of Proper Subsets of a Set A set of n elements has 2n – 1 proper subsets. Refer to the problems on the previous slide: 22  1 = 4  1 = 3 and 23  1 = 8  1 = 7. 11

Example: Given: A = { 12, 19, 26,…, 68} Find the number of subsets the number of proper subsets To solve this problem you need to determine how many elements are in the set. Notice the elements in this set are evenly spaced and form what is called an arithmetic sequence. The next term is found by adding a common difference of seven in this example. The total number of elements can be found without having to list all of the elements in this set using the formula An = A1 + (n – 1)d. An is the last term in the sequence, A1 is the first term, d is the common difference and n is the number of terms in the sequence. Substituting the numbers into the equation and solving for n yields the number of terms in this set. 68 = 12 + (n – 1)(7) –12 –12 56 = (n – 1)(7) 7 7 8 = n – 1 +1 + 1 9 = n a. Number of subsets = 29 = 512 b. Number of proper subsets = 29 – 1 = 512 – 1 = 511 12 END