Discrete Mathematics Assignment Help
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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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Q1. In this question, write down your answer, no need for any justification. You can leave your answer in terms of factorials , combination symbols, permutation symbols, etc. Please clearly box your answers in your submission to Gradescope.
 How many nonisomorphic (free) trees are there with 4 vertices?
 How many solutions are there to x _{1}+x _{2}+x _{3}+x _{4 }= 20 where x _{4 } ≥ 10 and each{" "} x_{i } 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{" "} B_{n } denote the number of partitions of a set with n elements and set B _{0 }= 1.
3. 3. Show that Bn satisfies the recurrence Bn .
(3 points) Show that for all n ≥ 1, 2 ^{n−} ^{1 } ≤ B_{n} .
(3 points) Show that for n ≥ 1,{" "} B_{n }≤{" "} 2 ^{n} ^{2}. 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{" "}
K_{n }
that have exactly n vertices (and prove that it
is correct)
 (5 points) Give a formula for the number of subgraphs of{" "} K_{n } (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 ncube have?
 (2 points) If the ncube was planar, how many faces would it have? (Your answer will depend on{" "} n).
 (4 points) Show that the 4cube (and hence every{" "} ncube 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{" "} e^{0 } in G have the same weight, then removing either e or{" "} e^{0 } 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 2
^{h }
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 T _{1 }and T _{2 }are each perfect binary trees of height{" "} h then T _{1 }and T _{2 }are isomorphic as binary trees.
 A 3ary 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 3ary tree is a 3ary tree where each parent has exactly 3 children.
Two 3ary trees are isomorphic as 3ary 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 3ary trees) 3ary trees with{" "} n vertices and the set of nonisomorphic (as 3ary trees) full 3ary trees with 2n + 1 terminal vertices.
 (5 points) Let{" "} t_{n } be the number of nonisomorphic (as 3ary trees) 3ary trees with n vertices, and by convention set{" "} t _{0 }= 1.
Show that t n i t it jt n−1−(i+j).