lect8 - Todays topics: The notion of a relation properties...

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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, …5} “less then” = { (1, 2), (1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5), (3, 4), (3, 5), (4, 5)}
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4 Let A = {1, 2, 3, 4}. Which ordered pairs are in the relation R = {( a, b ) | a divides b }? R = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 4), (3, 3), (4, 4)} A 1 2 3 4 A 1 2 3 4
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5 Consider the following relations on the set of integers Z: R 1 ={( a , b ) | a b } R 2 ={( a , b ) | a > b } R 3 ={( a , b ) | a = b or a = - b } R 4 ={( a , b ) | a = b +1 } R 5 ={( a , b ) | a + b 3 } Which of these relations contain each of the pairs (1, 1), (1, 2), (2, 1), (1, - 1) and (2, 2)?
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6 equal” = {( a , b )| a , b Power({1, 2, 3}) and | a| =| b | } “subset” ={( a , b )| a , b Power({1, 2, 3}) and a b } equal” = {( , ), ({1}, {1}), ({1}, {2}), ({2}, {1}), … ({1, 2}, {2, 3}), ({2, 3}, {1, 2}), …({1, 2, 3}, {1, 2, 3})} 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} subset” = {( , ), ( , {1}), ( , {2}), . .. ( , {1, 2, 3}), ({1}, {1}), ({1}, {1, 2}), ({1}, {1, 2, 3}), ({2}, {2}), … ({1, 2}, {1, 2}), ({1, 2}, {1, 2, 3}), ({1, 3}, {1, 3}), ({1, 3}, {1, 2, 3}), ({2, 3}, {1, 2, 3}), …({1, 2, 3}, {1, 2, 3})}
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7 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.
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lect8 - Todays topics: The notion of a relation properties...

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