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Comparing Sets. MATH 102 Contemporary Math S. Rook. Overview. Section 2.2 in the textbook: Set equality & set equivalence Subsets. Set Equality & Set Equivalence. Set Equality.

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comparing sets

Comparing Sets

MATH 102

Contemporary Math

S. Rook

overview
Overview
  • Section 2.2 in the textbook:
    • Set equality & set equivalence
    • Subsets
set equality
Set Equality
  • Two sets, A and B, are equal, denoted as A = B if they both contain exactly the same elements; otherwise, we write A ≠ B
    • Order DOES NOT matter
      • e.g. Let A = {1, 2, 3, 4, 5} and B = {5, 4, 3, 2, 1}. Does A = B?
    • If A = B, what can we say about n(A) and n(B)?
set equivalence
Set Equivalence
  • Two sets, A and B, are equivalent if n(A) = n(B)
    • i.e. the number of elements in each is the same
  • Set equality is NOT the same as set equivalence!!!
    • You must understand the difference!
  • e.g. Consider any finite set A
    • List the elements in set B so that AequalsB
    • List the elements in set B so that A is equivalent, but NOTequal to B
set equality example
Set Equality (Example)

Ex 1: Replace # with = or ≠ to make the statement true:

a) {2, 3, 5, 7} # {x | x is a prime number less than 12}

b) {y | y is a weekday} # {Friday, Monday, Thursday, Tuesday, Wednesday}

subsets1
Subsets
  • We say that A is a subset of B, denoted by if EVERY element of A is also in B
    • Again, order does NOT matter
      • e.g. Let A = {2, 6, 8, 10} and B = {14, 12, 10, 8, 6, 2}. Is A a subset of B?
    • If there is at least one element of A that is not in B, we write A n/s B
      • e.g. Consider sets A and B from above. Is B a subset of A?
subsets continued
Subsets (Continued)
  • Given sets A and B, if A is a subset of BANDA ≠ B, we say that A is a proper subset of B denoted
    • Note that BOTH conditions must be fulfilled for A to be a proper subset of B
    • e.g. Let A = {a, e, i, o, u} and B = { l | l is a letter of the alphabet }. Is A a proper subset of B?
    • e.g. Let A = {a, e, i, o, u} and B = {v | v is a vowel}. Is A a proper subset of B?
subsets example
Subsets (Example)

Ex 2: Replace the # with to make the statement true:

a) {t | t is a letter in the word ruth} # {z | z is a letter in the word truth}

b) Ø # {1, 2, 3, …, 100}

c) {Aberdeen, Darlington, Fallston} # {b | b is a building at HCC}

listing subsets
Listing Subsets
  • Sometimes it is useful to know all subsets of a set in order to assist in making decisions
    • See the options discussion on pg 49-50 of the textbook
  • For any set A:
    • The leastnumber of elements in A’s subsets is 0
      • How do we write a set with 0 elements?
    • The maximum number of elements in A’s subsets is n(A)
    • To list the subsets of A, we first list Ø and A and then list the subsets that have between 0 and n(A) elements
      • When n(B) = n(C), for any two subsets B and C of A, B ≠ C
        • i.e. Same-sized subsets must have different elements
      • e.g. Consider listing the subsets of A = {a, b}
listing subsets continued
Listing Subsets (Continued)
  • Now consider listing the subsets of B = {a, b, c}
    • What is the relationship between the number of subsets of a set with 2 elements versus a set with 3 elements?
      • The number of subsets of a set containing k elements is 2k
  • Consider again our subset listings for sets A and B
    • How many proper subsets are in each listing?
    • What is the relationship between the number of proper subsets of a set with 2 elements versus a set with 3 elements?
      • The number of proper subsets of a set containing k elements is 2k – 1
listing subsets example
Listing Subsets (Example)

Ex 3: The board of directors of a corporation own different amounts of stock which affects voting power. Adam has a voting power of 4, Beth has a voting power of 3, Chris 2, and Danielle 1. Any issue needs a voting weight of at least 6 to be passed. List all of the different possible voting combinations where an issue passes.

summary
Summary
  • After studying these slides, you should know how to do the following:
    • Given two sets A and B, determine whether they are equal or equivalent
    • Given two sets A and B, determine whether A is a subset, is not a subset, or is a proper subset of B
    • List all of the subsets of a given set A
  • Additional Practice:
    • See the list of suggested problems for 2.2
  • Next Lesson:
    • Set Operations (Section 2.3)