comparing sets math 102 contemporary math s. rook
TRANSCRIPT
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)