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# Discrete Mathematics

The study of mathematical structures that can be considered "discrete" rather than "continuous" and include integers, graphs, and statements

## and operating systems

In my safe is a sheet of paper with two shapes drawn on it in colored crayon. One is a square, and the other is a triangle. Each shape is drawn in a single color. Suppose you believe me when I tell you that if the square is blue, then the triangle is green. What do you therefore know about the truth value of the following statement? The square is not blue or the triangle is green.

• The statement is FALSE

Two edges are adjacent if they share a vertex.

• True
• False

The number of edges incident to a vertex.

• Degree of a vertex

Rule that states that every function can be described in four ways: algebraically (a formula), numerically (a table), graphically, or in words.

• Rule of four
• Rule of thumb
• Rule of function

De Morgan's law is used in finding the equivalence of a logic expression using other logical functions.

• True
• False

Does a rational r value for r2 =6 exist?

• No, a rational r does not exist.
• Yes, a rational r exist.

In a simple graph, the number of edges is equal to twice the sum of the degrees of the vertices.

• True
• False

Consider the statement, “If you will give me a cow, then I will give you magic beans.” Determine whether the statement below is the converse, the contrapositive, or neither. If you will give me a cow, then I will not give you magic beans.

• Converse

A function which renames the vertices.

• non-isomorphism
• isomorphism

The cardinality of {3, 5, 7, 9, 5} is 5.

• True
• False

Determine the number of elements in A U B.

• 18

A set of statements, one of which is called the conclusion and the rest of which are called premises.

• argument

Let A = {3, 4, 5}. Find the cardinality of P(A).

• 8

What is the 20th term?

• 29

How many possible output will be produced in a proposition of three statements?

• 8

A _____ is a _____ which starts and stops at the same vertex.

• Euler circuit, Euler path

When a connected graph can be drawn without any edges crossing, it is called ________________ .

• Edged graph
• Planar graph
• Spanning graph

Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed?

• 244000
• 2100
• 210
• 21100

A _____ is a function which is both an injection and surjection. In other words, if every element of the codomain is the image of exactly one element from the domain

• bijection

The geometric sequences uses common _____ in finding the succeeding terms.

• factor

All graphs have Euler's Path

• True
• False

Arithmetic progression is the sum of the terms of the arithmetic series.

• True
• False

Every connected graph has a spanning tree.

• True
• False

Find | R | when R = {2, 4, 6,..., 180}

• 90

If you travel to London by train, then the journey takes at least two hours.

• If your journey by train takes more than two hours, then you don't travel to London.
• If your journey by train takes less than two hours, then you don’t travel to London.

Let ‘G’ be a connected planar graph with 20 vertices and the degree of each vertex is 3. Find the number of regions in the graph.

• 12

Circuits start and stop at _______________

• different vertices
• same vertex

How many spanning trees are possible in the given figure?

• 4

Paths start and stop at the same vertex.

• True
• False

Indicate which, if any, of the following graphs G = (V, E, φ), |V | = 5, is not connected.

• φ = ( a {1,2} b {2,3} c {1,2} d {1,3} e {2,3} f {4,5} )
• φ = ( a {1,2} b {2,3} c {1,2} d {2,3} e {3,4} f {1,5} )
• φ = ( 1 {1,2} 2 {1,2} 3 {2,3} 4 {3,4} 5 {1,5} 6 {1,5} )

A Bipartite graph is a graph for which it is possible to divide the vertices into two disjoint sets such that there are no edges between any two vertices in the same set.

• True
• False

If the right angled triangle t, with sides of length a and b and hypotenuse of length c, has area equal to c2/4, what kind of triangle is this?

• obtuse triangle
• isosceles triangle
• scalene triangle

A statement which is true on the basis of its logical form alone.

• Tautology
• Double Negation
• De Morgan's Law

Find f (1).

• 4
• 1
• 3
• 2

Tracing all edges on a figure without picking up your pencil or repeating and starting and stopping at different spots

• Euler Path
• Euler Circuit

The study of what makes an argument good or bad.

• logic

How many simple non-isomorphic graphs are possible with 3 vertices?

• 4

Does this graph have an Euler Path, Euler Circuit, both, or neither?

• Euler Circuit
• None
• Euler Path
• Both

Let A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6, 7}

• {1, 2, 3, 5, 6, 7}
• {1, 2, 6, 7}
• {1, 2, 3, 4, 5, 6, 7}
• {3, 4, 5}

The sum of the geometric progression is called geometric series

• True
• False

What type of progression this suggest?

• Arithmetic

What is the 4th and 8th element of aNo= n^(2) ?

• 64,16
• 8,16
• 32,64
• 16,64

As soon as one vertex of a tree is designated as the _____, then every other vertex on the tree can be characterized by its position relative to the root.

• root

In how many different ways can the letters of the word 'OPTICAL' be arranged so that the vowels always come together?

• 720

How many people like apples only?

• 2

Identify the propositional logic of the truth table given

• disjunction
• negation
• conjunction
• implication

A sequence of vertices such that consecutive vertices (in the sequence) are adjacent (in the graph). A walk in which no edge is repeated is called a trail, and a trail in which no vertex is repeated (except possibly the first and last) is called a path

• Subgraph
• Walk
• Vertex coloring

A spanning tree that has the smallest possible combined weight.

• minimum spanning tree

Find |A ∩ B| when A = {1, 3, 5, 7, 9} and B {2, 4, 6, 8, 10}

• 0 (zero)

A graph in which every pair of vertices is adjacent.

• Complete Graph

Consider the statement, “If you will give me a cow, then I will give you magic beans.” Determine whether the statement below is the converse, the contrapositive, or neither. If I will give you magic beans, then you will give me a cow.

• Neither

Find the cardinality of R = {20,21,...,39, 40}

• 21

Deduction rule is an argument that is not always right.

• True
• False

How many 3-letter words with or without meaning, can be formed out of the letters of the word, 'LOGARITHMS', if repetition of letters is not allowed?

• 720

¬P ∨ Q is equivalent to :

• ¬(P∨Q)
• P → Q
• ¬P ∧ ¬Q

Additive principle states that if given two sets A and B, we have |A × B| |A| · |B|.

• True
• False

An argument form which is always valid.

• deduction rule

How many people like only one of the three?

• 26

What is the element n in the domain such as fNo = 1

• 2

How many people takes tea and wine?

• 32

It is a connected graph containing no cycles.

• Tree

A path which visits every vertex exactly once

• Hamilton Path

These are lines or curves that connect vertices.

• Edges

Two graphs that are the same are said to be _______________

• isomorphic
• isometric
• isochoric

What is the line covering number of for the following graph?

• 3

Indicate which, if any, of the following three graphs G = (V, E, φ), |V | = 5, is not isomorphic to any of the other two.

• φ = ( b {4,5} f {1,3} e {1,3} d {2,3} c {2,4} a {4,5} )
• φ = ( f {1,2} b {1,2} c {2,3} d {3,4} e {3,4} a {4,5} )
• φ = (A {1,3} B {2,4} C {1,2} D {2,3} E {3,5} F {4,5} )

A graph for which it is possible to divide the vertices into two disjoint sets such that there are no edges between any two vertices in the same set.

• Bipartite Graph

match the following formulas to its corresponding sequence

• Geometric Series, Double Summation

A graph T is a tree if and only if between every pair of distinct vertices of T there is a unique path.

• True
• False

For all n in rational, 1/n ≠ n - 1

• True
• False

A _____ graph has no isolated vertices.

• connected

A _____ graph has two distinct groups where no vertices in either group connecting to members of their own group

• bipartite

A tree is the same as a forest.

• True
• False

Find the cardinality of S = {1, {2,3,4},0} | S | = _____

• 3

An argument is said to be valid if the conclusion must be true whenever the premises are all true.

• True
• False

Which of the following is false?

• A graph with one odd vertex will have an Euler Path but not an Euler Circuit.
• Euler Paths exist when there are exactly two vertices of odd degree.
• A graph with more than two odd vertices will never have an Euler Path or Circuit.
• Euler circuits exist when the degree of all vertices are even

Find an element n of the domain such that f No = n.

• 3

_____ is a function from a subset of the set of integers.

• Sequence

A connected graph with no cycles.

• tree

What is the minimum height height of a full binary tree?

• 3

If n is a rational number, 1/n does not equal n-1.

• True
• False

Euler paths must touch all edges.

• True
• False

_____ is the same truth value under any assignment of truth values to their atomic parts.

• Logic Equivalence

The _____ is a subset of the codomain. It is the set of all elements which are assigned to at least one element of the domain by the function. That is, the range is the set of all outputs.

• range

Fill in the blanks. A graph F is a _____ if and only if between any pair of vertices in F there is at most _____

• forest, one path

If two vertices are adjacent, then we say one of them is the parent of the other, which is called the _____ of the parent.

• child

What is the matching number for the following graph?

• 4

How many people takes coffee but not tea and wine?

• 45

What is the missing term? 3,9,__,81....

• 27

Suppose P and Q are the statements: P: Jack passed math. Q: Jill passed math. Translate "¬(P ν Q) → Q" into English.

• Neither Jack or Jill passed math.
• Jill passed math if and only if Jack did not pass math.
• If Jack did not pass math and Jill did not pass math, then Jill did not pass math.
• If Jack or Jill did not pass math, then Jill passed math.

Match the following properties of trees to its definition.

• Proposition 4.2.1 → A graph T is a tree if and only if between every pair of distinct vertices of T there is a unique path., Proposition 4.2.4 → 4 Let T be a tree with v vertices and e edges. Then e v − 1., Corollary 4.2.2 → A graph F is a forest if and only if between any pair of vertices in F there is at most one path, Proposition 4.2.3 → Any tree with at least two vertices has at least two vertices of degree one.

Match the truth tables to its corresponding propositional logic

• Implication, Disjunction, Conjunction

The child of a child of a vertex is called

• grandchild

_____ is the simplest style of proof.

• Direct Proof

The minimum number of colors required in a proper vertex coloring of the graph.

• Chromatic number

A graph is complete if there is a path from any vertex to any other vertex.

• True
• False

How many edges would a complete graph have if it had 6 vertices?

• 30
• 20
• 25
• 15

The tree elements are called _____

• nodes

Which of the following statements is NOT TRUE?

• A graph F is a forest if and only if between any pair of vertices in F there is at most one path.
• Any tree with at least two vertices has at least two vertices of degree two.
• Let T be a tree with v vertices and e edges. Then e v − 1.

The given graph is planar.

• True
• False

What is the sum from 1st to 5th element?

• 40

¬(P ∨ Q) is logically equal to which of the following expressions?

• ¬P ∧ ¬Q.
• ¬P ∨ Q
• ¬P ∨ ¬Q

The number of simple digraphs with |V | = 3 is

• 512

Consider the statement, “If you will give me a cow, then I will give you magic beans.” Determine whether the statement below is the converse, the contrapositive, or neither. If you will not give me a cow, then I will not give you magic beans.

• Converse

It is a rule that assigns each input exactly one output

• function

Consider the function f : N → N given by f (0) 0 and f (n + 1) f No + 2n + 1. Find f (6).

• 36

Consider the statement, “If you will give me a cow, then I will give you magic beans.” Determine whether the statement below is the converse, the contrapositive, or neither. If I will not give you magic beans, then you will not give me a cow.

• Contrapositive

In my safe is a sheet of paper with two shapes drawn on it in colored crayon. One is a square, and the other is a triangle. Each shape is drawn in a single color. Suppose you believe me when I tell you that if the square is blue, then the triangle is green. What do you therefore know about the truth value of the following statement? If the triangle is green, then the square is blue.

• The statement is TRUE

A sequence of vertices such that every vertex in the sequence is adjacent to the vertices before and after it in the sequence

• walk

What is the difference of persons who take wine and coffee to the persons who the persons who takes tea only?

• 15

An undirected graph G which is connected and acyclic is called ____________.

• forest
• cyclic graph
• tree
• bipartite graph

Defined as the product of all the whole numbers from 1 to n.

• factorial

Solve for the value of n in :

• -31

surjective and injecive are opposites of each other.

• True
• False

Which of the following the logic representation of proof by contrapositive?

• P → Q = ¬Q → P
• P → Q = Q → ¬P
• P → Q = ¬Q → ¬P
• P → Q = ¬(Q → P)

Which of the following is a possible range of the function?

• All numbers except 3
• 1,2,3
• 3,6,9,12 only
• all multiples of three
• 3,4,5,6,7,8,9,10

A simple graph has no loops nor multiple edges.

• True
• False

What is the type of progression?

• Arithmetic

IN combinations, the arrangement of the elements is in a specific order.

• True
• False

A graph is an ordered pair G (V, E) consisting of a nonempty set V (called the vertices) and a set E (called the edges) of two-element subsets of V.

• True
• False

Proofs that is used when statements cannot be rephrased as implications.

The _____ of a a subset B of the codomain is the set f −1 (B) {x ∈ X : f (x) ∈ B}.

• inverse image

It is an algorithm for traversing or searching tree or graph data structures.

• depth first search.
• spanning tree

A sequence that involves a common difference in identifying the succeeding terms.

• Geometric Progression
• Arithmetic Progression

Consider the statement, “If you will give me a cow, then I will give you magic beans.” Determine whether the statement below is the converse, the contrapositive, or neither. You will give me a cow and I will not give you magic beans.

• Contrapositive

A _____ connected graph with no cycles. (If we remove the requirement that the graph is connected, the graph is called a forest.) The vertices in a tree with degree 1 are called _____

• tree
• tree
• leaves

A sequence of vertices such that consecutive vertices (in the sequence) are adjacent (in the graph). A walk in which no edge is repeated is called a trail, and a trail in which no vertex is repeated (except possibly the first and last) is called a path.

• Walk

The ________________________ states that if event A can occur in m ways, and event B can occur in n disjoint ways, then the event “A or B” can occur in m + n ways.