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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

How many people like apples only?

• 2

For all n in rational, 1/n ≠ n - 1

• True
• False

The number of edges incident to a vertex.

• Degree of a vertex

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

• walk

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

• 8

Two graphs that are the same are said to be _______________

• isomorphic
• isometric
• isochoric

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.

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

• True
• False

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

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

• argument

The child of a child of a vertex is called

• grandchild

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

• 15

The tree elements are called _____

• nodes

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

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

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

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

• 40

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

• Sequence

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

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 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

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

• factor

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

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} )

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

• depth first search.
• spanning tree

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

• 720

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

A _____ graph has no isolated vertices.

• connected

Identify the propositional logic of the truth table given

• disjunction
• negation
• conjunction
• implication

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

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

• Geometric Progression
• Arithmetic Progression

The study of what makes an argument good or bad.

• logic

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.

Determine the number of elements in A U B.

• 18

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

A graph in which every pair of vertices is adjacent.

• Complete Graph

Deduction rule is an argument that is not always right.

• True
• False

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

• Tautology
• Double Negation
• De Morgan's Law

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

• 3

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

• 3

Circuits start and stop at _______________

• different vertices
• same vertex

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

• 3

What is the 20th term?

• 29

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}

How many people takes tea and wine?

• 32

A spanning tree that has the smallest possible combined weight.

• minimum spanning tree

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

A path which visits every vertex exactly once

• Hamilton Path

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)

A connected graph with no cycles.

• tree

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

• 8

All graphs have Euler's Path

• True
• False

These are lines or curves that connect vertices.

• Edges

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

• 0 (zero)

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

How many people takes coffee but not tea and wine?

• 45

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.

What type of progression this suggest?

• Arithmetic

The sum of the geometric progression is called geometric series

• True
• False

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

• 64,16
• 8,16
• 32,64
• 16,64

What is the type of progression?

• Arithmetic

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

• True
• False

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

• True
• False

Find f (1).

• 4
• 1
• 3
• 2

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

• 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 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

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

• ¬P ∧ ¬Q.
• ¬P ∨ Q
• ¬P ∨ ¬Q

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

• True
• False

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

• 90

A function which renames the vertices.

• non-isomorphism
• isomorphism

Paths start and stop at the same vertex.

• True
• False

Two edges are adjacent if they share a vertex.

• True
• False

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

• Euler circuit, Euler path

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

• forest
• cyclic graph
• tree
• bipartite graph

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

• True
• False

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

• 36

match the following formulas to its corresponding sequence

• Geometric Series, Double Summation

How many spanning trees are possible in the given figure?

• 4

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

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 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

A simple graph has no loops nor multiple edges.

• True
• False

How many people like only one of the three?

• 26

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

• 30
• 20
• 25
• 15

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

• Euler Circuit
• None
• Euler Path
• Both

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

• 2

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

• bipartite

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

• True
• False

_____ is the simplest style of proof.

• Direct Proof

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

• Edged graph
• Planar graph
• Spanning graph

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 number of simple digraphs with |V | = 3 is

• 512

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

• True
• False

surjective and injecive are opposites of each other.

• True
• False

It is a rule that assigns each input exactly one output

• function

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

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

• inverse image

Every connected graph has a spanning tree.

• True
• False

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

• 27

It is a connected graph containing no cycles.

• Tree

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

What is the matching number for the following graph?

• 4

Solve for the value of n in :

• -31

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

• 3

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

• 4

Euler paths must touch all edges.

• True
• False

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

• 21

The given graph is planar.

• True
• False

Match the truth tables to its corresponding propositional logic

• Implication, Disjunction, Conjunction

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

• 244000
• 2100
• 210
• 21100

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

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

• Chromatic number

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

• Logic Equivalence

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

Does a rational r value for r2 =6 exist?

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

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

• Euler Path
• Euler Circuit

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

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

¬P ∨ Q is equivalent to :

• ¬(P∨Q)
• P → Q
• ¬P ∧ ¬Q

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

• True
• False

A tree is the same as a forest.

• True
• False

An argument form which is always valid.

• deduction rule

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.

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

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.