# sol3 - MACM 101 — Discrete Mathematics I Outline...

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MACM 101 — Discrete Mathematics I Outline Solutions to Exercises on Functions and Relations 1. Make a list of pairs, construct the matrix, and draw the graph of the relation R from the set A = { 0 , 1 , 2 , 3 , 4 } to the set B = { 0 , 1 , 2 , 3 } such that ( a,b ) R if and only if a + b = 4 . The set of paier R = { (1 , 3) , (2 , 2) , (3 , 1) , (4 , 0) } ; the matrix 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 , and the graph 0 1 2 3 4 0 1 2 3 B A 2. Prove that ( A B ) × C = ( A × C ) ( B × C ) . Method 1. We have ( A B ) × C = { ( a,b ) | a A B b C } = { ( a,b ) | ( a A a B ) b C } = { ( a,b ) | ( a A b C ) ( a B b C ) } = { ( a,b ) | a A b C } ∪ { ( a,b ) | ( a B b C ) } = ( A × C ) ( B × C ) . Method 2. We show that ( A B ) × C ( A × C ) ( B × C ) , and that ( A × C ) ( B × C ) ( A B ) × C . ( A B ) × C ( A × C ) ( B × C ) . Take an element ( a,b ) from ( A B ) × C . Then a A B , and hence a A or a B . Since b C , in the former case we have ( a,b ) A × C , wand in the latter case we have ( a,b ) B × C . In either case ( a,b ) ( A × C ) ( B × C ) . ( A

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sol3 - MACM 101 — Discrete Mathematics I Outline...

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