# Set and Set Operations

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1 Set and Set Operations

2 Introduction A set is a collection of objects. The objects in a set are called elements of the set. A well defined set is a set in which we know for sure if an element belongs to that set. Example: The set of all movies in which John Cazale appears is well defined. (Name the movies, and what do they have in common? There are only 5.) The set of all movie serials made by Republic Pictures is well defined. The set of best TV shows of all time is not well defined. (It is a matter of opinion.)

3 Ways to define sets Explicitly: {John, Paul, George, Ringo} Implicitly: {1,2,3, }, or {2,3,5,7,11,13,17, } Set builder: { x : x is prime }, { x x is odd }. In general { x : P(x)}, where P(x) is some predicate. We read the set of all x such that P(x)

4 Set Builder Notation When it is not convenient to list all the elements of a set, we use a notation the employs the rules in which an element is a member of the set. This is called set builder notation. V = { people citizens registered to vote in Maricopa County} A = {x x > 5} = This is the set A that has all real numbers greater than 5. The symbol is read as such that.

5 Set properties 1 Order does not matter We often write them in order because it is easier for humans to understand it that way {1, 2, 3, 4, 5} is equivalent to {3, 5, 2, 4, 1} Sets are notated with curly brackets 5

6 Set properties 2 Sets do not have duplicate elements Consider the set of vowels in the alphabet. It makes no sense to list them as {a, a, a, e, i, o, o, o, o, o, u} What we really want is just {a, e, i, o, u} Consider the list of students in this class Again, it does not make sense to list somebody twice Note that a list is like a set, but order does matter and duplicate elements are allowed We won t be studying lists much in this class 6

7 Specifying a set 1 Sets are usually represented by a capital letter (A, B, S, etc.) Elements are usually represented by an italic lower-case letter (a, x, y, etc.) Easiest way to specify a set is to list all the elements: A = {1, 2, 3, 4, 5} Not always feasible for large or infinite sets 7

8 Specifying a set 2 Can use an ellipsis ( ): B = {0, 1, 2, 3, } Can cause confusion. Consider the set C = {3, 5, 7, }. What comes next? If the set is all odd integers greater than 2, it is 9 If the set is all prime numbers greater than 2, it is 11 Can use set-builder notation D = {x x is prime and x > 2} E = {x x is odd and x > 2} The vertical bar means such that Thus, set D is read (in English) as: all elements x such that x is prime and x is greater than 2 8

9 Specifying a set 3 A set is said to contain the various members or elements that make up the set If an element a is a member of (or an element of) a set S, we use then notation a S 4 {1, 2, 3, 4} If an element is not a member of (or an element of) a set S, we use the notation a S 7 {1, 2, 3, 4} Virginia {1, 2, 3, 4} 9

10 Special Sets of Numbers N = The set of natural numbers. = {1, 2, 3, }. W = The set of whole numbers. ={0, 1, 2, 3, } Z = The set of integers. = {, -3, -2, -1, 0, 1, 2, 3, } Q = The set of rational numbers. ={x x=p/q, where p and q are elements of Z and q 0 } H = The set of irrational numbers. R = The set of real numbers. C = The set of complex numbers.

11 Universal Set and Subsets The Universal Set denoted by U is the set of all possible elements used in a problem. When every element of one set is also an element of another set, we say the first set is a subset. Example A={1, 2, 3, 4, 5} and B={2, 3} We say that B is a subset of A. The notation we use is B A. Let S={1,2,3}, list all the subsets of S. The subsets of S are, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}.

12 The Empty Set The empty set is a special set. It contains no elements. It is usually denoted as { } or. The empty set is always considered a subset of any set. Do not be confused by this question: Is this set {0} empty? It is not empty! It contains the element zero.

13 Intersection of sets When an element of a set belongs to two or more sets we say the sets will intersect. The intersection of a set A and a set B is denoted by A B. A B = {x x is in A and x is in B} Note the usage of and. This is similar to conjunction. A ^ B. Example A={1, 3, 5, 7, 9} and B={1, 2, 3, 4, 5} Then A B = {1, 3, 5}. Note that 1, 3, 5 are in both A and B.

14 Mutually Exclusive Sets We say two sets A and B are mutually exclusive if A B =. Think of this as two events that can not happen at the same time.

15 Union of sets The union of two sets A, B is denoted by A U B. A U B = {x x is in A or x is in B} Note the usage of or. This is similar to disjunction A v B. Using the set A and the set B from the previous slide, then the union of A, B is A U B = {1, 2, 3, 4, 5, 7, 9}. The elements of the union are in A or in B or in both. If elements are in both sets, we do not repeat them.

16 Complement of a Set The complement of set A is denoted by A or by A C. A = {x x is not in set A}. The complement set operation is analogous to the negation operation in logic. Example Say U={1,2,3,4,5}, A={1,2}, then A = {3,4,5}.

17 Cardinal Number The Cardinal Number of a set is the number of elements in the set and is denoted by n(a). Let A={2,4,6,8,10}, then n(a)=5. The Cardinal Number formula for the union of two sets is n(a U B)=n(A) + n(b) n(a B). The Cardinal number formula for the complement of a set is n(a) + n(a )=n(u).

18 Examples. {1, 2, 3} is the set containing 1 and 2 and 3. {1, 1, 2, 3, 3} = {1, 2, 3} since repetition is irrelevant. {1, 2, 3} = {3, 2, 1} since sets are unordered. {0,1, 2, 3, } is a way we denote an infinite set (in this case, the natural numbers). = {} is the empty set, or the set containing no element. Note: { }

19 Definitions and notation x S means x is an element of set S. x S means x is not an element of set S. A B means A is a subset of B. or, B contains A. or, every element of A is also in B. or, x ((x A) (x B)). A B Venn Diagram

20 Definitions and notation A B means A is a subset of B. A B means A is a superset of B. A = B if and only if A and B have exactly the same elements iff, A B and B A iff, A B and A B iff, x ((x A) (x B)). So to show equality of sets A and B, show: A B B A

21 Definitions and notation A B means A is a proper subset of B. A B, and A B. x ((x A) (x B)) x ((x B) (x A)) A B

22 Definitions and notation Quick examples: {1,2,3} {1,2,3,4,5} {1,2,3} {1,2,3,4,5} Is {1,2,3}? Is {1,2,3}? Is {,1,2,3}? Is {,1,2,3}? Yes! x (x ) (x {1,2,3}) holds, because (x ) is false. No! Yes! Yes!

23 Cardinality If S is finite, then the cardinality of S, S, is the number of distinct elements in S. If S = {1,2,3} If S = {3,3,3,3,3} S = 3. S = 1. If S = S = 0. If S = {, { }, {,{ }} } S = 3. If S = {0,1,2,3, }, S is infinite. (more on this later)

24 Power sets If S is a set, then the power set of S is P(S) = 2 S = { x : x S }. If S = {a} 2 S = {, {a}}. We say, P(S) is the set of all subsets of S. If S = {a,b} If S = 2 S = { }. 2 S = {, {a}, {b}, {a,b}}. If S = {,{ }} 2 S = {, { }, {{ }}, {,{ }}}. Fact: if S is finite, 2 S = 2 S. (if S = n, 2 S = 2 n )

25 Cartesian Product The Cartesian Product of two sets A and B is: A B = { (a, b) : a A b B} If A = {Charlie, Lucy, Linus}, and B = {Brown, VanPelt}, then A B = {(Charlie, Brown), (Lucy, Brown), (Linus, Brown), (Charlie, VanPelt), (Lucy, VanPelt), (Linus, VanPelt)} We ll use these special sets soon! A 1 A 2 A n = = {(a 1, a 2,, a n ): a 1 A 1, a 2 A 2,, a n A n } A,B finite A B = A B

26 Set Operations

27 union The union of two sets A and B is: A B = { x : x A x B} If A = {Charlie, Lucy, Linus}, and B = {Lucy, Desi}, then A B = {Charlie, Lucy, Linus, Desi} B A

28 intersection The intersection of two sets A and B is: A B = { x : x A x B} If A = {Charlie, Lucy, Linus}, and B = {Lucy, Desi}, then A B = {Lucy} B A

29 The intersection of two sets A and B is: A B = { x : x A x B} If A = {x : x is a US president}, and B = {x : x is in this room}, then A B = {x : x is a US president in this room} = B A Sets whose intersection is empty are called disjoint sets

30 complement The complement of a set A is: If A = {x : x is not shaded}, then U A = U and U =

31 symmetric difference The symmetric difference, A B, is: A B = { x : (x A x B) (x B x A)} = (A B) (B A) = { x : x A x B} U A B B A

32 Set Identities Identity Domination Idempotent A U = A A = A A U = U A = A A = A A A = A

33 2.2 Set Identities Excluded Middle Uniqueness Double complement

34 Set Identities Commutativity Associativity C) Distributivity C) C) A B = B A A B = B A (A B) C = A (B (A B) C =A (B C) A (B C) = (A B) (A A (B C) = (A B) (A

35 Subsets 1 If all the elements of a set S are also elements of a set T, then S is a subset of T For example, if S = {2, 4, 6} and T = {1, 2, 3, 4, 5, 6, 7}, then S is a subset of T This is specified by S T Or by {2, 4, 6} {1, 2, 3, 4, 5, 6, 7} If S is not a subset of T, it is written as such: S T For example, {1, 2, 8} {1, 2, 3, 4, 5, 6, 7} 35

36 Subsets 2 Note that any set is a subset of itself! Given set S = {2, 4, 6}, since all the elements of S are elements of S, S is a subset of itself This is kind of like saying 5 is less than or equal to 5 Thus, for any set S, S S 36

37 Subsets 3 The empty set is a subset of all sets (including itself!) Recall that all sets are subsets of themselves All sets are subsets of the universal set A horrible way to define a subset: x ( x A x B ) English translation: for all possible values of x, (meaning for all possible elements of a set), if x is an element of A, then x is an element of B This type of notation will be gone over later 37

38 Proper Subsets 1 If S is a subset of T, and S is not equal to T, then S is a proper subset of T Let T = {0, 1, 2, 3, 4, 5} If S = {1, 2, 3}, S is not equal to T, and S is a subset of T A proper subset is written as S T Let R = {0, 1, 2, 3, 4, 5}. R is equal to T, and thus is a subset (but not a proper subset) or T Can be written as: R T and R T (or just R = T) Let Q = {4, 5, 6}. Q is neither a subset or T nor a proper subset of T 38

39 Proper Subsets 2 The difference between subset and proper subset is like the difference between less than or equal to and less than for numbers The empty set is a proper subset of all sets other than the empty set (as it is equal to the empty set) 39

40 Proper subsets: Venn diagram S R R U S 40

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