## CS6105 Discrete Mathematics ( Final Exam )

The course covers the basic concepts and techniques of discrete mathematics. The course includes the discussion of mathematical logic, propositions, quantifiers, predicates, proof techniques, mathematical induction, fundamentals of set theory, sets, power sets, algebra of sets, relations, functions, count ability and finiteness, graphs and trees.

### Question 1

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Let A, B and C represent people who like apples, bananas, and carrots respectively. The number of people in A = 10, B = 12 and C = 16. Three people are such that they enjoy apples, bananas as well as carrots. Two of them like apples and bananas. Let three people like apples and carrots. Also, four people are such that they like bananas and carrots.

How many people like apples only?

How many people like only one of the three?

### Question 2

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

| R | =

### Question 3

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

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### Question 4

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A tree is the same as a forest.

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### Question 5

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For a function f : N → N, a recursive definition consists of an initial condition   together with a recurrence relation .

### Question 6

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

### Question 7

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Two graphs that are the same are said to be _______________

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### Question 8

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Determine whether the sentence below is an atomic statement, a molecular statement, or not a statement at all.

• The customers wore shoes.

### Question 9

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¬P ∨ Q is equivalent to :

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### Question 10

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Two edges are adjacent if they share a vertex.

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### Question 11

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Paths start and stop at the same vertex.

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### Question 12

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

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### Question 13

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Fill in the blanks.

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

### Question 14

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Classify the sentence below as an atomic statement, a molecular statement, or not a statement at all. If the statement is molecular, identify what kind it is (conjuction, disjunction, conditional, biconditional, negation).

We can have donuts for dinner, but only if it rains.

### Question 15

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

### Question 16

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### Question 17

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Euler paths must touch all edges.

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### Question 18

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

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### Question 19

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Assume the sequence: 1,3,5,7,9, ….

What is the 20th term?

What type of progression this suggest?

### Question 20

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

### Question 21

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is the simplest style of proof.

### Question 22

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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.

### Question 23

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Solve for the value of n in :

### Question 24

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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  .

### Question 25

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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.

### Question 26

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match the following formulas to its corresponding sequence Answer 1Choose...Geometric SeriesDouble SummationArithmetic Series Answer 2Choose...Geometric SeriesDouble SummationArithmetic Series

### Question 27

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

Find A ∩ B

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### Question 28

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What is the missing term?

3,9,__,81....

### Question 29

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surjective and injecive are opposites  of each other.

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### Question 30

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These are lines or curves that connect vertices.

### Question 31

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

### Question 32

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

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### Question 33

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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.

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### Question 34

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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 and the triangle are both green.

### Question 35

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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.

### Question 36

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graph has no isolated vertices.

### Question 37

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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

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### Question 38

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

### Question 39

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

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### Question 40

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Suppose P and Q are the statements:

P: Jack passed math.

Q: Jill passed math.

Translate "¬(P ν Q) → Q" into English.

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### Question 41

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### Question 42

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

Find A \ B

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### Question 43

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### Question 44

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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?

### Question 45

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Match the following properties of trees to its definition.
 Proposition 4.2.1 Answer 1Choose...Any tree with at least two vertices has at least two vertices of degree one.4 Let T be a tree with v vertices and e edges. Then e  v − 1.A graph F is a forest if and only if between any pair of vertices in F there is at most one pathA graph T is a tree if and only if between every pair of distinct vertices of T there is a unique path. Corollary 4.2.2 Answer 2Choose...Any tree with at least two vertices has at least two vertices of degree one.4 Let T be a tree with v vertices and e edges. Then e  v − 1.A graph F is a forest if and only if between any pair of vertices in F there is at most one pathA graph T is a tree if and only if between every pair of distinct vertices of T there is a unique path. Proposition 4.2.3 Answer 3Choose...Any tree with at least two vertices has at least two vertices of degree one.4 Let T be a tree with v vertices and e edges. Then e  v − 1.A graph F is a forest if and only if between any pair of vertices in F there is at most one pathA 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 Answer 4Choose...Any tree with at least two vertices has at least two vertices of degree one.4 Let T be a tree with v vertices and e edges. Then e  v − 1.A graph F is a forest if and only if between any pair of vertices in F there is at most one pathA graph T is a tree if and only if between every pair of distinct vertices of T there is a unique path.

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### Question 47

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

Find A U B

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### Question 48

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### Question 49

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

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