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Discrete Mathematics is the branch of Mathematics dealing with objects that can assume only distinct, separated values. The term "Discrete Mathematics" is therefore used in contrast with "Continuous Mathematics," which is the branch of Mathematics dealing with objects that can vary smoothly (and which includes, for example, calculus). Whereas Discrete objects can often be characterized by Integers, continuous objects require Real Numbers.

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The study of how discrete objects combine with one another and the probabilities of various outcomes is known as combinatorics. Other fields of Mathematics that are considered to be part of Discrete Mathematics include Graph Theory and the theory of computation. Topics in Number Theory such as congruences and recurrence relations are also considered part of Discrete Mathematics. ---reference: DiscreteMathematics

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### Sample of Discrete Maths Assignment Help Solved by the Experts

```Q1. In this question, write down your answer, no need for any justification. You can leave your answer in terms of factorials
```
• How many nonisomorphic (free) trees are there with 4 vertices?
• How many solutions are there to x1+x2+x3+x4 = 20 where x4 10 and each xi is a natural number (0 counts as a natural number for us).
• Let X = {1,2,3,4,5}. How many relations are there on X with the property that for all x ∈ X, x is not related to itself?
• Give the equivalence relation on {a,b,c,d,e} whose equivalence classes give the partition P = {{a,b},{c,d},{e}}.
• You are dividing 112 apples among 5 boxes. What is the smallest number of apples that could appear in the box with the most apples?
`2. Let X be a set with n elements. Be sure to justify all your answers.`
• (3 points) How many reflexive relations are there on X?
• (4 points) How many antisymmetric relations are there on X?
• (3 points) How many reflexive or antisymmetric relations are there on X?
• (4 points) Let Bn denote the number of partitions of a set with n elements and set B0 = 1.
`3. 3. Show that Bn satisfies the recurrence Bn .`

(3 points) Show that for all n ≥ 1, 2n−1 ≤ Bn.

(3 points) Show that for n ≥ 1, Bn 2n2. Hint: Remember that for a nonempty set there is a bijection between partitions on that set and equivalence relations on that set.

• (a) (5 points) Give a formula for the number of subgraphs of Kn that have exactly n vertices (and prove that it is correct)
• (5 points) Give a formula for the number of subgraphs of Kn (and prove that it is correct).
• (a) (2 points) Show that every cycle in the n-cube is of length 4 or longer.
• (2 points) How many edges does the n-cube have?
• (2 points) If the n-cube was planar, how many faces would it have? (Your answer will depend on n).
• (4 points) Show that the 4-cube (and hence every n-cube with n ≥ 4) is not planar.
• (a) (5 points) Suppose that G is a simple connected graph with finitely many vertices, and suppose that e is an edge in G such that removing e from G results in a disconnected graph. Show that e is in every spanning tree of G.
• (5 points) Suppose that G is a simple connected weighted graph with finitely many vertices and that if distinct edges e and e0 in G have the same weight, then removing either e or e0 from G results in a disconnected graph.

Show that G has a unique minimal spanning tree.

• A perfect binary tree of height h is a binary tree of height h with 2h terminal vertices.
• (5 points) Show that if T is a perfect binary tree of height h then the left and right subtrees of the root are each perfect binary trees of height h −
• (5 points) Show that if T1 and T2 are each perfect binary trees of height h then T1 and T2 are isomorphic as binary trees.
• A 3-ary tree is a rooted tree where each parent has at most three children, and each child is labeled with 1, 2, or 3 (and siblings all have different labels). A full 3-ary tree is a 3-ary tree where each parent has exactly 3 children.

Two 3-ary trees are isomorphic as 3-ary trees if they are isomorphic as rooted trees and the isomorphism preserves the labels of the children.

• (5 points) Show that there is a bijection between the set of nonisomorphic (as 3-ary trees) 3-ary trees with n vertices and the set of nonisomorphic (as 3-ary trees) full 3-ary trees with 2n + 1 terminal vertices.
• (5 points) Let tn be the number of nonisomorphic (as 3-ary trees) 3-ary trees with n vertices, and by convention set t0 = 1.

Show that tn i titjtn−1(i+j)).

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