Write an assignment about Discrete Structures in Computer Science

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Prove that if G′ is a solution to the minimum network design problem then G′ is a tree

 

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1. Prove, using the induction on the number of elements n of set A, that there exactly kn functions from A  into a set B with exactly k elements

 

2. Let A be an n element set and B be a k element set with k < n. Prove the following:

(a) There is onto mapping from A to B.

(b) There is an injective mapping from B to A.

 

3. Let G = (V, E) be a weighted connected graph. Consider graphs of the form G′ = (V, E′ ) where E′ ⊆ E. The cost of such subgraph G′ is the sum of the weights of the edges in E′ . We want to build a subgraph G′ = (V, E′ ) such that G0 is connected and the cost of G′ is as minimum as possible. We call this the minimum network design problem.

Prove that if G′ is a solution to the minimum network design problem then G′ is a tree

 

4. Let B be the set of all strings over the alphabet {0, 1, 2}. Consider the function f : BB such that for any string x, the value f(x) is obtained by replacing all 0s in x by 1s, and all 1s in x by 2s, and all 2s in x by 0s. Is the function bijective? Explain your answer.

 

5. Consider the function f : Z² → Z²  defined as follows. For (x, y) ∈ Z² : if both x and y are even or both x and y are odd then f(x, y) = (x, y + 1); otherwise f(x, y) = (x + 1, y).

Prove that f is a bijective function.

 

6. Let f : ZZ be the function defined as: f(n) = (−1)n + n. Prove that f is a bijective function.

 

7. Let A and B be finite sets. Prove that if f : AB is a bijection, then A and B have the same number of elements

 

8. Assume that there is a path in graph G from vertex x to vertex y, where xy. Prove that there is a path from x to y such that no vertex in this path is repeated.

 

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