**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**.

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*is a natural number (0 counts as a natural number for us)._{i } - 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*denote the number of partitions of a set with_{n }*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

- (a) (5 points) Give a formula for the number of subgraphs of
*K*that have exactly_{n }*n*vertices (and prove that it is correct)- (5 points) Give a formula for the number of subgraphs of
*K*(and prove that it is correct)._{n }

- (5 points) Give a formula for the number of subgraphs of
- (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.

- (2 points) How many edges does the

- (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*in^{0 }*G*have the same weight, then removing either*e*or*e*from^{0 }*G*results in a disconnected graph.

- (5 points) Suppose that

Show that *G *has a unique minimal spanning tree.

- A perfect binary tree of height
*h*is a binary tree of height*h*with 2terminal vertices.^{h }- (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.

- (5 points) Show that if
- 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 2*n*+ 1 terminal vertices. - (5 points) Let
*t*be the number of nonisomorphic (as 3-ary trees) 3-ary trees with_{n }*n*vertices, and by convention set*t*_{0 }= 1.

Show that *t**n **i t**it**jt**n−*1*−*(*i*+*j*)).

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