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Math Chapter 6 Part II. POWER SETS. In mathematics, given a set S , the power set of S , written P ( S ) or 2 n( S) , is the set of all subsets of S .
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POWER SETS • In mathematics, given a set S, the power set of S, written P(S) or 2n(S), is the set of all subsets of S. • Remember the definition of a subset is as follows: Given sets S and T, then T is defined to be a subset of S if every element of T is also an element of S.
POWER SETS • If S is the set {a, b, c} then {a,c} is a subset of S. There are other subsets of S; the complete list is as follows: • {} (the empty set or null set) • {a} • {b} • {c} • {a, b} • {a, c} • {b, c} • {a, b, c} or S • So the power set of S, written P(S), is the set containing all the subsets above. Written out this would be the set: • P(S) = { {}, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} }
POWER SETS • The power set of S, written P(S) or 2n(S), is the set of all subsets of S. • 2n(S) : given that a set S is finite then P(S) contains 2 raised to the number of elements is set S • S={1,2,3} • 23 = 8 subsets in the power set
PARTITIONS • A partition of set S is a subdivision of S into non-overlapping, nonempty subsets. • A partition of S is a collection of nonempty subsets such that: • Each element in S belongs to one of the subsets (all elements must be accounted for) • The subsets must be disjointed (no duplications of elements in the subsets)
PARTITIONS • S = {1,3,5,7,9} • Valid partitions: • [{1,3,5} {7,9}] • [{1,5,9} {3, 7}] • All elements are accounted for and there are no duplications • Invalid partitions: • [{1,5} {7, 9}] (3 is missing) • [{1,5,9} {1, 3, 7}] (duplication)
ORDERED PAIRS • An ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element. • An ordered pair with first element a and second element b is usually written as (a, b). • Two such ordered pairs (a1, b1) and (a2, b2) are equal if and only if a1 = a2 and b1 = b2.
ORDERED PAIRS • The set of all ordered pairs whose first element is in some set X and second element in some set Y is called the Cartesian product of said sets. • Given two setsX and Y, the Cartesian product (or direct product) of the two sets, written as X × Y (read as X cross Y) is the set of all ordered pairs with the first element of each pair selected from X and the second element selected from Y. • X × Y = { (x,y) | x in X and y in Y }
ORDERED PAIRS • For example: • set X is the 13-element set {A, K, Q, J, 10, 9, 8, 7, 6, 5, 4, 3, 2} • set Y is the 4-element set {spades, hearts, diamonds, clubs} • the Cartesian product of those two sets is: • the 52-element set { (A, spades), (K, spades), ... ,(2, spades), (A, hearts), ... , (3, clubs), (2, clubs) }.
RELATIONS • A binary relation is a mathematical concept to do with "relations", such as "is greater than" and "is equal to" in arithmetic, or "is an element of" in set theory. • Formally, a binary relation over a setX and a set Y is a ordered triple R=(X, Y, G(R)) where G(R), called the graph of the relation R, is a subset of X × Y. If (x,y) ∈ G(R) then we say that x is R-related to y and write xRy or R(x,y).
RELATIONS • Example: Suppose there are four objects: {ball, car, doll, gun} and four persons: {John, Mary, Sue, Venus}. Suppose that John owns the ball, Mary owns the doll, and Venus owns the car. No one owns the gun and Sue owns nothing. Then the binary relation "is owned by" is given as • R=({ball, car, doll, gun}, {John, Mary, Sue, Venus}, {(ball,John), (doll,Mary), (car,Venus)}). • The pair (ball,John), denoted by ballRJohn means ball is owned by John.
RELATIONS • It may also be thought of as a binary function that takes as arguments an element x of X and an element y of Y and evaluates to true or false (indicating whether the ordered pair (x, y) is an element of the set which is the relation).
PICTORIAL REPRESENTATION OF RELATIONS • Draw the coordinate diagram of A X B • Use a rectangular array whose rows are labeled by the elements of A and whose columns are labeled by the elements of B. • Write down the elements of A and the elements of B in two disjoint disks and then draw an arrow from the element in A the is related to the element in B.
PICTORIAL REPRESENTATION OF RELATIONS • Draw the coordinate diagram of A X B • set A = {1,2,3} set B = {a,b} • R = [{(1,a), (1,b), (3,a)}] R is the relation from set A to set B b a 1 2 3
PICTORIAL REPRESENTATION OF RELATIONS • Use a rectangular array whose rows are labeled by the elements of A and whose columns are labeled by the elements of B.
PICTORIAL REPRESENTATION OF RELATIONS • Write down the elements of A and the elements of B in two disjoint disks and then draw an arrow from the element in A the is related to the element in B. 1 2 3 a b
EQUIVALENCE RELATIONS • An equivalence relation on a setX is a binary relation on X that is reflexive, symmetric and transitive, i.e., if the relation is written as ~ it holds for all a, b and c in X that • (Reflexivity) a ~ a • (Symmetry) if a ~ b then b ~ a • (Transitivity) if a ~ b and b ~ c then a ~ c • Equivalence relations are often used to group together objects that are similar in some sense.
EQUIVALENCE RELATIONS • Partitioning into equivalence classes • Every equivalence relation on X defines a partition of X into subsets called equivalence classes: • all elements equivalent to each other are put into one class. Conversely, if the set X can be partitioned into subsets, then we can define an equivalence relation ~ on X by the rule "a ~ b if and only if a and b lie in the same subset". • If an equivalence relation ~ on X is given, then the set of all its equivalence classes is the quotient set of X by ~ and is denoted by X/~.
FUNCTIONS • The concept of function is a generalization of the common notion of a "mathematicalformula". Functions describe special mathematical relationships between two objects, x and y=f(x). The object x is called the argument of the function f, and y is said to "depend functionally" on x.
FUNCTIONS • Formal Definition • Formally, a function f from a set X of input values to a set Y of possibly output values (written as f: X → Y) is a relation between X and Y which satisfies: • 1. f is functional: if x f y (x is f-related to y) and x f z, then y = z. i.e., for each input value, there should only be one possible output value. • 2. f is total: for all x in X, there exists a y in Y such that x f y. i.e. for each input value, the formula should produce at least one output value within Y. • For each input value x in the domain, the corresponding unique output value y in the codomain is denoted by f(x).
FUNCTIONS • Domains, Codomains, and Ranges • X, the set of input values, is called the domain of f and Y, the set of possible output values, is called the codomain. • The range of f is the set of all actual outputs {f(x) : x in the domain}. Beware that sometimes the codomain is wrongly called the range because of a failure to distinguish between possible and actual values.
FUNCTIONS • Graph of a functions • The graph of a functionf is the collection of all points(x, f(x)), for all x in set X. In the example of the discrete function, the graph of f is {(1,a),(2,d),(3,c)}. There are theorems formulated or proved most easily in terms of the graph, such as the closed graph theorem.
FUNCTIONS • If X and Y are real lines, then this definition coincides with the familiar sense of graph. Here is the graph of a cubic function, which is a curve: • Note that since a relation on the two sets X and Y is usually formalized as a subset of X×Y, the formal definition of function actually identifies the function f with its graph.
