1. Comparing Sets MATH 102 Contemporary Math S. Rook

2. Overview Section 2.2 in the textbook: – Set equality & set equivalence – Subsets

3. Set Equality & Set Equivalence

4. 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)?

5. 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

6. 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}

7. Subsets

8. 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?

9. 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?

10. 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}

11. 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}

12. 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 2 k 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 2 k – 1

13. 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.

14. 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)