self-test - Fall 2009 CS131 – Combinatorial Structures...

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Unformatted text preview: Fall 2009 CS131 – Combinatorial Structures Self-test questions Self-test questions These questions are provided for additional self-testing, on material for which the book does not have enough self-test questions. Problem 1. Let A = { a , b , c }, B = { b , d }, C = {1,2}. What are the elements of the set ( A \ B ) × C ? Solution. These are pairs ( x , y ) with x ∈ A \ B = { a , c } and y ∈ C , so they are ( a ,1),( a ,2),( c ,1),( c ,2). Problem 2. Show A × ( B ∪ C ) = ( A × B ) ∪ ( A × C ). Solution. Look at a pair ( x , y ). It is in A × ( B ∪ C ) if and only if x ∈ A and y ∈ B ∪ C . This holds if and only if x ∈ A , and either y ∈ B or y ∈ C . This holds if and only if either x ∈ A , and y ∈ B or x ∈ A and y ∈ C . This holds if and only if either x ∈ A × B or x ∈ A × C . This holds if and only if x ∈ ( A × B ) ∪ ( A × C ). Problem 3. Let Z = X × Y be a Cartesian product, and let A ⊆ X and B ⊆ Y . Show that then for the indicator function I A × B ( x , y ) we have the equality I A × B ( x , y ) = I A ( x ) · I B ( y ). Solution. We have I A × B ( x , y ) = 1 if and only if ( x , y ) ∈ A × B . This holds if and only if x ∈ A and y ∈ B . This holds if and only if I A ( x ) = 1 and I B ( y ) = 1. This holds if and only if I A ( x ) · I B ( y ) = 1. Problem 4. Let A = {0,1,2,..., a- 1}, and let f : A → A be defined by f ( x ) = 2 x mod a . For what values of a is this function injective? For what values is it surjective? Solution. Since A is finite, the function either both injective and surjective or nei- ther. Now, if a is even then the sequence f (0), f (1), f (2),..., f ( a- 2 2 ), f a 2 , f a + 2 2 ,..., f ( a- 1) is 0,2,4,..., a- 2,0,2,..., a- 2, so f is not injective. If a is odd then the sequence f (0), f (1), f (2),..., f a- 1 2 , f a + 1 2 , f a + 3 2 ,..., f ( a- 1) is 0,2,4,..., a- 1,1,3,..., a- 2, so f is injective. Fall 2009 CS131 – Combinatorial Structures Self-test questions Problem 5.Problem 5....
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self-test - Fall 2009 CS131 – Combinatorial Structures...

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