Discrete Mathematics Section 5 02 Subsets ST 1

Discrete Mathematics Section 5. 02 -Subsets ST 1

Objective: The students will… Distinguish between a subset and a proper subset.

Subsets Large sets are often broken into smaller more manageable sets, called subsets. Subset: Set A is a subset of set B, symbolized by A B, if and only if all the elements of set A are also elements of set B.

Subsets Example Set B is the people in your class. A subset of this could be any of the following: * females * students whose last name begins with M * students who are 17 year olds

Example Is N a subset of M? M = {5, 7, 9, 11, 13, 15}, N = {5, 7, 13} Answer: Yes! Every element of set N is also found inside set M, so N M.

You Try Is set A a subset of set B? A = {2, 3, 4, 5}, B = {2, 3} Answer: No! Set A has a 4 and a 5 that were not found inside set B, so A B. A is not a subset of B

You Try Write two sets, A and B, so that A B. Possible Answer: A = {1, 2, 3} and B = {e, f, g} In this case: A B

Proper Subset: Set A is a proper subset of set B, symbolized by A B, if and only if all the elements of set A are elements of set B and Set A Set B (all subsets are proper subsets unless they are exactly the same)

Proper Subset Example Is set A a proper subset of set B if A={jazz, pop, hip hop}, B={classical, jazz, pop, rap, hip hop}? Answer: Yes. Every element in Set A is found inside Set B but they are not identical, thus A B.

Example If M = {1, 2, 3, 4}, N = {2, 3, 4} and O = {1, 2, 3, 4}, are N and O proper subsets of M? Answer: N is a proper subset of M, so N M O is not a proper subset of M, so O M, but O M

Important: Empty Set The empty set is a proper subset of every set except itself. Ex: If A = { } and B = { }, then A is NOT a proper subset of B (because they are identical) BUT: the empty set is a subset of every set including itself so, A B but A B

Number of Distinct Subsets The “number of distinct subsets” of a finite set A is , where n is the number of elements in set A.

Distinct Subsets Example: If A = {a, b, c} what is the number of distinct subsets? What are they? Answer: so there are 8 subsets. The subsets are: {a, b, c}, {a, b}, {a, c}, {b, c}, {a}, {b}, {c}, { }

You Try At the local pizza shop, you can order pizza with pepperoni, sausage, Canadian bacon, olives, or anchovies. How many different types of pizza can be made? Answer: types of pizza

Lesson Reminders: When dealing with identical sets, remember… ◦ Every set is a subset of itself: A A ◦ No set is a proper subset of itself: A A

Assignment Worksheet 5. 02