Comparing Sets

MATH 102Contemporary Math

S. Rook

Overview

• Section 2.2 in the textbook:– Set equality & set equivalence– Subsets

Set Equality & Set Equivalence

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

• 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 A equals B– List the elements in set B so that A is equivalent,

but NOT equal to B

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}

Subsets

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?

BA

Subsets (Continued)

• Given sets A and B, if A is a subset of B AND A ≠ 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?

BA

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}

n/sor , ,

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 least number 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)• 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)

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• 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)