Which of the following sentences is a proposition? a)The smallest...

Which of the following sentences is a proposition?

a)The smallest perfect number is 12

b)When is the final exam?

c)Do your homework

d)What a beautiful day!

Let p be "It is Wednesday" and q be "Our CMSC 150 class is held today", what is the proper English translation for p q?

a)It is not Wednesday, and our CMSC 150 class is held today

b)It is Wednesday, and our CMSC 150 class is held today

c)Our CMSC 150 class is held every Wednesday

d)It is Wednesday, but or CMSC 150 class is not held today

When is the proposition p (p q)) false

a)When both p and q are true

b)When both p and q are false

c)When p is true and q is false

d)When p is false and q is true

Which of the following correctly expresses the fact that is commutative?

a) q q p

b) (p q) p q

c) ( q) r p (q r)

d) (q r) (p q) (p r)

Which of the following describes the proposition p p (p p))?

a) It is a contradiction

b) It is both a tautology and a contradiction

c) It is neither a tautology nor a contradiction

d) It is a tautology

Which of the following expressions is the negation of the expression: x > 5 or < 10?

a) 5 or y 10

b) 5 and 10

c) >5 and y < 10

d) < 5 or > 10

What is the contrapositive of proposition: p q r?

a) q r

b) q r

c) q r p

d) q r p

Which of the following describes modus ponens?

a) , p q q

b) q, p q p

c) p, p q q

d) q, p q p

In the predicate calculus statement "n , n > m", which of the following correctly describes the variables n and m?

a)n is a bound variable and m is a free variable

b)Both n and m are free variables

c)n is a free variable and m is a bound variable

d)Both n and m are bound variables

Which of the following is the correct predicate calculus translation of the sentence "Some natural numbers are at least 100"?

a) n , n > 100

b) n , n > 100

c) n , n 100

d) n , n 100

Which of the following is the negation of the predicate calculus statement "n , n > 0 n < 0"?

a) n , n > 0 n < 0

b) n , n 0 n < 0

c) n , n 0 n 0

d) n , n 0 n 0

Which of the following is the negation of the following sentence "Everyone love chocolate"?

a)Everyone doesn't love chocolate

b)Someone loves chocolate

c)No one loves chocolate

d)At least one person doesn't love chocolate

Which of the following counterexamples demonstrates that the following statement of predicate calculus is false: "n , m , nm > 0 (n > 0 m > 0)"?

a)n = 1 and m = 1

b)n = -1 and m = 1

c)n = 0 and m = 0

d)n = -1 and m = -1

Which of the following predicate calculus statements is true?

a) n , m , n + 1 = m

b) n , m , n = m + 1

c) n , m , n + 1 = m

d) n , m , n + 1 = m

Suppose that in proving that the sum of two odd numbers is even, we begin by saying "Assume that the sum of two odd numbers is odd". Which of the following types of proof are we using?

a)Direct proof

b)Proof by contraposition

c)Proof by cases

d)Proof by contradiction

Which of the following real number counterexamples provides a disproof of the universal statement "x , 10x x"?

a)x =10

b)x = -10

c)x = 0

d)x = 5

In the proof by induction of the theorem n , n 1 2 + 4 + ... + 2n = n + n, which of the following would be the proper way to state the base case?

a) When n = 2, the sum consists of the first term 2 and the formula shows 1 + 1 = 2

b) When n = 0, the sum consists of the first term 0 and the formula shows 0 + 0 = 0

c) When n = 1, the sum consists of the first term 1 and the formula shows 1 = 1

d) When n = 1, the sum consists of the first term 2 and the formula shows 1 + 1 = 2

Which of the following represents the set of all even positive integers less than 10?

a){0,2,4,6,8}

b){2,4,6,8,10}

c){2,4,6,8}

d){0,2,4,6,8,10}

Among the sets {a,b,c},{a,b},c} and {a,b,c}, which has the highest cardinality?

a){a,b,c}

b){a,b,c}

c)All three have the same cardinality

d){a,b}c}

Given an alphabet = {a, b} and a language L consisting of all strings of length 3 that begin with a, which of the following is the set of all strings in L?

a){aaa,aab,aba,abb}

b){a,aa,ab,aaa,abb}

c){a,aa,ab,aaa,aab,aba,abb}

d){a,aa,ab}

Assume A is the set of positive integers less than 10 and B is the set of positive integers less than or equal to 20, and R is a relation from A to B defined as follows: R = {(a, b) | a A, b B, a is divisible by 4 b = 2a}. Which of the following ordered pairs belongs to that relation?

a)(16,32)

b)(8,16)

c)(12,6)

d)(6,12)

Given the relation R = {(n, m) | n, m , n 0 m 0}. Among reflexive, symmetric, antisymmetric and transitive, which of those properties are true of this relation?

a)It is only transitive

b)It is both reflexive and symmetric

c)It is both antisymmetric and transitive

d)It is both symmetric and transitive

Given the relation R = {(n, m) | n, m , n/3 = m/3}. Which of the following is one of the equivalence classes of this relation?

a){1,3,5,7}

b){1,2,3}

c){-3,-2,-1,0,1,2,3}

d){3,4,5}

Which of the following is a synonym for a one-to-one function?

a)surjection

b)rejection

c)bijection

d)injection

Which of the following functions is an onto function?

a)f: , where f(x) = 2x + 1

b)f: , where f(n) = 4n + 2

c)f: , where f(n) = |3n|

d)f: , where f(x) = 2x + 7

Given functions f: , where f(x) = 2x + 3, and g: , where g(x) = x + 5, which of the following functions is fg?

a)fg: , where fg(x) = 4x + 14

b)fg: , where fg(x) = 2x + 13

c)fg: , where fg(x) = 2x + 8

d)fg: , where fg(x) = 4x + 12x + 14

Which of the following two pairs of sets have the same cardinality?

a)Natural Numbers and Irrational Numbers

b)Real Numbers and Rational Numbers

c)Irrational Numbers and Rational Numbers

d)Integers and Natural Numbers

How many odd integers are there between 100 and 200 inclusive?

a)100

b)101

c)50

d)51

If you throw a pair of six-sided diced with the faces numbered 1 to 6, what is the probability that the sum of the two faces adds to 5?

a)1/9

b)1/6

c)1/18

d)1/3

How many anagrams are there for the word root, including the word itself?

a)18

b)12

c)8

d)4

Suppose 37 students are enrolling in one of four courses. There will be at least one course that has at least how many students?

a)28

b)46

c)10

d)9

Two vertices of a graph are adjacent when which of the following is true?

a)Both vertices have even degrees

b)There is an edge that is between them

c)Both vertices are isolated

d)There is a path of length 2 that connects them

Which of the following best describes a complete graph?

a)All the vertices of the graph are isolated

b)Every vertex in the graph has a odd degree

c)The graph contains a edge between every pair of vertices

d)The sum of all the degrees of every vertex is odd

Which of the following is true about spanning trees of the above graph?

a)It has only one possible spanning tree

b)It has no possible spanning tree because it contains an isolated vertex

c)It has many possible spanning trees

d)It has no possible spanning tree because it is disconnected