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CS 441 Discrete mathematics for CS^ M. Hauskrecht
CS 441 Discrete Mathematics for CS
Lecture 7
Milos Hauskrecht [email protected] 5329 Sennott Square
Sets and set operations
CS 441 Discrete mathematics for CS^ M. Hauskrecht
Basic discrete structures
- Discrete math =
- study of the discrete structures used to represent discrete objects
- Many discrete structures are built using sets
- Sets = collection of objects
Examples of discrete structures built with the help of sets:
- Combinations
- Relations
- Graphs
CS 441 Discrete mathematics for CS^ M. Hauskrecht
Set
- Definition : A set is a (unordered) collection of objects. These objects are sometimes called elements or members of the set. (Cantor's naive definition)
- Examples:
- Vowels in the English alphabet V = { a, e, i, o, u }
- First seven prime numbers. X = { 2, 3, 5, 7, 11, 13, 17 }
CS 441 Discrete mathematics for CS^ M. Hauskrecht
Representing sets
**Representing a set by:
- Listing (enumerating) the members of the set.
- Definition by property, using the set builder notation** {x| x has property P}. Example:
- Even integers between 50 and 63. 1) E = {50, 52, 54, 56, 58, 60, 62} 2) E = {x| 50 <= x < 63, x is an even integer}
If enumeration of the members is hard we often use ellipses. Example: a set of integers between 1 and 100
CS 441 Discrete mathematics for CS^ M. Hauskrecht
Equality
Definition: Two sets are equal if and only if they have the same elements.
Example:
- {1,2,3} = {3,1,2} = {1,2,1,3,2}
Note: Duplicates don't contribute anything new to a set, so remove them. The order of the elements in a set doesn't contribute anything new.
Example: Are {1,2,3,4} and {1,2,2,4} equal? No!
CS 441 Discrete mathematics for CS^ M. Hauskrecht
Special sets
- Special sets:
- The universal set is denoted by U: the set of all objects under the consideration.
- The empty set is denoted as or { }.
CS 441 Discrete mathematics for CS^ M. Hauskrecht
Venn diagrams
- A set can be visualized using Venn Diagrams :
U A B C
CS 441 Discrete mathematics for CS^ M. Hauskrecht
A Subset
- Definition : A set A is said to be a subset of B if and only if every element of A is also an element of B. We use A B to indicate A is a subset of B.
- Alternate way to define A is a subset of B: x (x A) (x B)
U
A
B
CS 441 Discrete mathematics for CS^ M. Hauskrecht
A proper subset
Definition : A set A is said to be a proper subset of B if and only if A B and A B. We denote that A is a proper subset of B with the notation A B. U
A
B
CS 441 Discrete mathematics for CS^ M. Hauskrecht
A proper subset
Definition : A set A is said to be a proper subset of B if and only if A B and A B. We denote that A is a proper subset of B with the notation A B.
Example: A={1,2,3} B ={1,2,3,4,5} Is: A B? Yes.
U
A
B
CS 441 Discrete mathematics for CS^ M. Hauskrecht
Cardinality
Definition: Let S be a set. If there are exactly n distinct elements in S, where n is a nonnegative integer, we say S is a finite set and that n is the cardinality of S. The cardinality of S is denoted by | S |.
Examples:
• A={1,2,3,4, …, 20}
|A| =
CS 441 Discrete mathematics for CS^ M. Hauskrecht
Infinite set
Definition : A set is infinite if it is not finite.
Examples:
- The set of natural numbers is an infinite set.
- N = {1, 2, 3, ... }
- The set of reals is an infinite set.
CS 441 Discrete mathematics for CS^ M. Hauskrecht
Power set
• P( {1} ) = { , {1} }
• |P({1})| = 2
- Assume {1,2}
- P( {1,2} ) = { , {1}, {2}, {1,2} }
- |P({1,2} )| =
- Assume {1,2,3}
- P({1,2,3}) = {, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} }
- |P({1,2,3} | = 8
- If S is a set with |S| = n then | P(S) | = 2 n
CS 441 Discrete mathematics for CS^ M. Hauskrecht
N-tuple
- Sets are used to represent unordered collections.
- Ordered-n tuples are used to represent an ordered collection.
Definition : An ordered n-tuple (x1, x2, ..., xN) is the ordered collection that has x1 as its first element, x2 as its second element, ..., and xN as its N-th element, N 2.
Example:
- Coordinates of a point in the 2-D plane (12, 16)
x
y
CS 441 Discrete mathematics for CS^ M. Hauskrecht
Cartesian product
Definition : Let S and T be sets. The Cartesian product of S and T , denoted by S x T, is the set of all ordered pairs (s,t), where s S and t T. Hence,
- S x T = { (s,t) | s S t T}.
Examples:
- S = {1,2} and T = {a,b,c}
- S x T = { (1,a), (1,b), (1,c), (2,a), (2,b), (2,c) }
- T x S = { (a,1), (a, 2), (b,1), (b,2), (c,1), (c,2) }
- Note: S x T T x S !!!!
CS 441 Discrete mathematics for CS^ M. Hauskrecht
Cardinality of the Cartesian product
Example:
- A= {John, Peter, Mike}
- B ={Jane, Ann, Laura}
- A x B= {(John, Jane),(John, Ann) , (John, Laura), (Peter, Jane), (Peter, Ann) , (Peter, Laura) , (Mike, Jane) , (Mike, Ann) , (Mike, Laura)}
- |A x B| = 9
- |A|=3, |B|=3 |A| |B|= 9
Definition: A subset of the Cartesian product A x B is called a relation from the set A to the set B.
CS 441 Discrete mathematics for CS^ M. Hauskrecht
Disjoint sets
Definition : Two sets are called disjoint if their intersection is empty.
- Alternate: A and B are disjoint if and only if A B = .
Example:
- A={1,2,3,6} B={4,7,8} Are these disjoint?
- Yes.
- A B =
U B A
CS 441 Discrete mathematics for CS^ M. Hauskrecht
Cardinality of the set union
Cardinality of the set union.
- |A B| = |A| + |B| - |A B|
- Why this formula?
^ U A
B
CS 441 Discrete mathematics for CS^ M. Hauskrecht
Cardinality of the set union
Cardinality of the set union.
- |A B| = |A| + |B| - |A B|
- Why this formula? Correct for an over-count.
- More general rule:
- The principle of inclusion and exclusion.
^ U A
B
CS 441 Discrete mathematics for CS^ M. Hauskrecht
Set difference
Definition : Let A and B be sets. The difference of A and B , denoted by A - B , is the set containing those elements that are in A but not in B. The difference of A and B is also called the complement of B with respect to A.
- Alternate: A - B = { x | x A x B }.
Example: A= {1,2,3,5,7} B = {1,5,6,8}
U A
B
