lect8_1 - Todays topics: The notion of a relation...

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1 The notion of a relation properties of relations on a set Today’s topics: Relations A “ relation ” is a fundamental mathematical notion expressing a relationship between sets It’s an abstract notion useful for modeling many different relationships
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2 Example . Let S be set of UCF students, and C be a set of classes. Then we can consider the relation “is taking class” from S to C S C x y This relation can be described by the set of pairs: “is taking class”={( x , y )| x S , y C and student x is taking class y }
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3 More examples of relations: “parent-of” “child-of” “likes” “meet one another today” “less then” = {( a , b ) | a , b A and a < b } where A ={1, 2, …10} equal” = {( a , b )| a , b Power( A ) and | a| =| b | } “subset” ={( a , b )| a , b Power( A ) and a b } If R is set of real numbers, R × R is set of points ( x , y ) in plane. “circle”={( x , y )| x , y R and x 2 + y 2 =1}
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4 You don’t need to give a meaningful name to a relation. The only thing that really matters about relations is that we know which elements in A are related to which element of B. A relation R is completely described if we know R -related pairs Suppose A ={1, 2, 3}, B ={ r , s } and we know 1 Rr , 2 Rs , 3 Rr , then we know everything we need to know about R. R can be specified by the list of pairs: R ={(1, r ), (2, s ), (3, r )} The Cartesian product contains all possible pairs: A × B = {(1, r ), (1, s ), (2, r ), (2, s ), (3, r ), (3, s )}
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5 Definition . A binary relation from A to B is a subset R A × B. Conversely, any subset of A × B can be considered as a relation.
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lect8_1 - Todays topics: The notion of a relation...

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