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Notes-Sets - Sets Relations Functions and Infinity Dr David...

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Sets, Relations, Functions and Infinity Dr. David A. Workman School of EE and CS Tuesday, May 16, 2006
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Sets Sets are unordered collections of distinct objects. Sets can be defined or specified in many ways: By explicitly enumerating their members or elements e.g. S = { a, b, c} Note: If S' = { b, c, a}, then S and S' denote the same set (that is, S' = S) By specifying a condition for membership S = { x | P(x) }, reads "S is the set of all x in such that P(x) is true" P is called a "predicate" ( a function from set to {true, false} ) E.g. S = { x UCF | x is a CS major } The empty set is denoted, Φ , and is the set with no members; that is, Φ = { }. Also, the predicate, x Φ , is always false ! If S Φ , then there exists an x for which x S is true; this predicate is read " x is an element of S " or " x is a member of S ". The symbol " " denotes the member relation . If A and B are sets, then we write " A B " to mean that A is a subset of B . This means that for all x A, x B. Or, 2200 x [x A x B]. The expression, " A B " means that A is a proper subset of B . Mathematically, 2200 x [x A x B] and 5 y [ y B and y A] The cross product of two sets A and B is denoted, A × B , and is the set defined as follows: A × B = { (a,b) | a A and b B } . "(a,b)" is an expression composed from elements, a,b, selected arbitrarily from sets A and B, respectively. If A B, then A × B B × A. In fact, Exercise: Prove A × B = B × A if and only if A = B.
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Relations A relation, r, is a mapping from some set A to some set B; We write, r: A B, and we mean that r assigns to every member of A a subset of B; that is, for every a A, r(a) B and r(a) Φ . A relation, r, can also be defined in terms of the cross product of A and B: r A × B such that for every a A there is b B such that (a, b) r. A relation can be graphed as illustrated by the example below. Example: Let A = { a, b, c} , B = { 0, 1, 2 }, and r = { (a,0), (a,2), (b,1), (c,0) } r(a) = { 0, 2 } r(b) = { 1 } r(c) = { 0 } B A a b c 2 1 0 Graph of relation, r We say that a relation, r, from A to B is a partial relation if and only if for some a A, r(a) = Φ .
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Functions Functions are special types of relations. Specifically, a relation f: A B, is said to be a (total) function from A to B if and only if, for every a A, f(a) has exactly one element ; that is, |f(a)| = 1. If f is a partial function from A to B , then f may not be defined for every a A. In this case we write |f(a)| 1, for every a in A; note that |f(a)| = 0 if and only if f(a) = Φ , and we say the function is undefined at a. A function, f, is said to be one-to-one ( 1-1) if and only if x y implies f(x) f(y). A total function that is one-to-one is sometimes called an injection.
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