20 Slides--More on Relations

20 Slides--More on Relations - CS103 HO#20 Slides-More on...

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CS103 HO#20 Slides--More on Relations April 19, 2010 1 A binary relation on set A is a set of ordered pairs from A × A. kid age a 4 b 8 c 4 d 7 e 8 f 4 If S is the relation Same Age , then S includes If R is the relation Older Than , then R includes a b c d e f b c d e f a A A binary relation on set A is a set of ordered pairs from A × A. kid age a 4 b 8 c 4 d 7 e 8 f 4 If S is the relation Same Age , then S includes If R is the relation Older Than , then R includes a b c d e f b c d e f a Reflexive x A(xSx) Irreflexive x A(¬xRx) A binary relation on set A is a set of ordered pairs from A × A. kid age a 4 b 8 c 4 d 7 e 8 f 4 If S is the relation Same Age , then S includes If R is the relation Older Than , then R includes a b c d e f b c d e f a Reflexive x A(xSx) Symmetric x,y A(xSy ySx) Irreflexive x A(¬xRx) Antisymmetric x,y A((xRy x y) ¬yRx) A binary relation on set A is a set of ordered pairs from A × A. kid age a 4 b 8 c 4 d 7 e 8 f 4 If S is the relation Same Age , then S includes If R is the relation Older Than , then R includes a b c d e f b c d e f a Reflexive x A(xSx) Symmetric x,y A(xSy ySx) Transitive x,y,z A((xRy yRz) xRz) Irreflexive x A(¬xSx) Antisymmetric x,y A((xRy x y) ¬yRx) Transitive x,y,z A((xRy yRz) xRz) b c d e f a Same Age b c d e f a Same Age
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CS103 HO#20 Slides--More on Relations April 19, 2010 2 b c d e f a Same Age A partition of a set S is a collection of disjoint, non-empty subsets of S that have S as their union. A relation R on S that is reflexive, symmetric, and transitive is called an equivalence relation on S, and R partitions S into a collection of disjoint equivalence classes . b c d e f a R is an Equivalence Relation on S If R is an equivalence relation on a set S, and a S, then the set of all elements x such that aRx is called the equivalence class of a and is denoted by [a] R . If only one relation is being discussed, we will omit the subscript R. Thus [a] = {x | (a, x) R} For the set shown above, [a] = [f] = [c]. The individual elements are called representatives of the same equivalence class {a, f, c}. b c d e f a R is an Equivalence Relation on S If R is an equivalence relation on a set S, and a S, then the set of all elements x such that aRx is called the equivalence class of a and is denoted by [a] R . If only one relation is being discussed, we will omit the subscript R. Thus [a] = {x | (a, x) R} The set of sets { [a] | a S } is a partition of A.
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20 Slides--More on Relations - CS103 HO#20 Slides-More on...

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