MAD2104 HW 2.1-2.3 +practice test

Match the Veen diagram to the corresponding operations with sets.

(A-B)∪(A-C)∪(B-C)

Let A = {0, 2, 4, 6, 8, 10}, B = {0, 1, 2, 3, 4, 5, 6}, and C = {4, 5, 6, 7, 8, 9, 10}. Find A∪ B∪ C

{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Let A = {0, 2, 4, 6, 8, 10}, B = {0, 1, 2, 3, 4, 5, 6}, and C = {4, 5, 6, 7, 8, 9, 10}. Find (A∩ B) ∪ C

{0, 2, 4, 5, 6, 7, 8, 9, 10}

Let Ai= {1, 2, 3, 4..., i} where i = 1, 2, 3, ... Find ∩= Intersection of AI, i=1,.,n

{1}

Let A = {0, 2, 4, 6, 8, 10}, B = {0, 1, 2, 3, 4, 5, 6}, and C = {4, 5, 6, 7, 8, 9, 10}. Find (A ∪ B) ∩ C

{4, 5, 6, 8, 10}

Let A = {0, 2, 4, 6, 8, 10}, B = {0, 1, 2, 3, 4, 5, 6}, and C = {4, 5, 6, 7, 8, 9, 10}. Find A∩ B∩ C.

{4, 6}

Find the Truth Set for the predicate P(x): x3≥ 1 if the domain is the set of integers

{x ∈Z | x ≥ 1}

Describe the set {−3, −2, −1, 0, 1, 2, 3} using Set-Builder notation

{x∈ Z | −3 ≤ x ≤ 3}

Find the Truth Set for the predicate P(x): x2= 2 if the domain is the set of integers

Ø

Find the inverse of the function f(x)=ax+b

f^-1(x) = (x - b) /a

Given the graph of the function f, for which values of x is f(x)=0?

x=-3, x=2, x=4

Let A= {a, b, c, d} and B = {y, z}. Find B × A.

{(y, a), (y, b), (y, c), (y, d), (z, a), (z, b), (z, c), (z, d)}

Match the Veen diagram to the corresponding operations with sets.

(A ∩ C) ∪ (B ∩ A)

Suppose that the universal set is U= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Express the set {3, 4, 5} with a bit string where the ith bit in the string is 1 if i is in the set and 0 otherwise.

00 1110 0000

Determine the cardinality of the set A = {Ø}

1 because it is a set with exactly one element, the empty set.

Calculate E^15J=8 with cj=5(3)^j

107 600 400

Initially, a pendulum swings through an arc of 15 inches. See the figure. On each successive swing, the length of the arc is 0.98 of the previous length.. After 10 swings, what total distance will the pendulum have swung?

137.2 inches

Initially, a pendulum swings through an arc of 15 inches. See the figure. On each successive swing, the length of the arc is 0.98 of the previous length. What is the length of the arc of the 10th swing?

15 (0.98)9

Find the cardinality of the power set P({Ø, a, {a}, { {a} } })

16

Calculate ∑^2i=0∑^3j=0i2j3

180

Find the sum of the finite geometric series: 3, 6, 12, . . . , 1536.

3069

Find the next 3 terms of the recursive formula defined by a1=4, an=2an-1+5

4,13,31,67

Initially, a pendulum swings through an arc of 15 inches. See the figure. On each successive swing, the length of the arc is 0.98 of the previous When it stops, what total distance will the pendulum have swung?

750 inches

Find two sets A and B such that A ∈ B and A ⊆ B.

A = ∅ and B = {∅}

Suppose that A= {2, 4, 6}, B = {2, 6, 7}, C = {4, 6}, and D = {4, 6, 8}. Identify the pairs of sets in which one is a subset of the other (in any order). (Check all that apply.)

A, C

Suppose A× B = Ø, where A and B are sets. What can you conclude?

A= Ø or B = Ø as A × B will be nonempty if A and B are nonempty.

Find the sets A and B if A − B = {1, 5, 8}, B − A = {2, 10}, and A ∩ B = {3, 9}.

A=(1,3,5,8,9), B=(2,3,9,10)

Let f(x) = x2+ 1 and g(x) = x + 2 be functions from R to R. If f(g(x)) = x2 + Ax + B and g(f(x)) = x2 + Cx + D, the corresponding values of A,B,C and D are

A=4, B=5, C=0, D=3

Suppose A× B = Ø, where A and B are sets. What can you conclude?

A≠ Ø and B ≠ Ø as product can be Ø if none of the sets are empty.

if A, B, and C are sets such that A ∪ C = B ∪ C, we can conclude that A = B.

False

if A, B, and Care sets such that A ∩ C = B ∩ C, we can conclude that A = B.

False

A relation on a nonempty set A is called a partial ordering or a partial-order relation if it is reflexive, antisymmetric, and transitive. Which of the following relations is not a partial-order relation?

In the set of all integers the relation x is related to y if the remainder of the division of x and y by 4 is the same.

If cardinality of (M U Q) is equal to (cardinality of M + cardinality of Q), this means that

M and Q are disjoint

Assume that the population of the world in 2010 was 6.5 billion and is growing at the rate of 1.1% a year. Find an explicit formula for the population of the world n years after 2010.

Pn = (1.1)n (6,5) billion

Assume that the population of the world in 2010 was 6.5 billion and is growing at the rate of 1.1% a year.. Set up a recurrence relation for the population of the world n years after 2010.

Pn+1= 1.1 Pn, P0= 6,5 billion

An employee joined a company in 2009 with a starting salary of $50,000. Every year this employee receives a raise of $1000 plus 5% of the salary of the previous year. Set up a recurrence relation for the salary of this employee n years after 2009.

Sn+1= (1.05)Sn + 1000, S0=50000

An employee joined a company in 2009 with a starting salary of $50,000. Every year this employee receives a raise of $1000 plus 5% of the salary of the previous year. Find an explicit formula for the salary of this employee n years after 2009.

Sn=(1.05)n(5000)+[(1.05)n-1]/0.05

Decide whether the given relation on the set {1, 2, 3, 4}, is reflexive, symmetric, antisymmetric, or transitive. {(2, 4), (4, 2)}

Symmetric

Identify whether the following collection of subsets is a partition of S= {−3,−2,−1, 0, 1, 2, 3} and the correct reason for it {−3,−2,−1, 0}, {0, 1, 2, 3}

The given collection of sets does not form a partition of S as these sets are not mutually disjoint.

Identify whether the following collections of subsets are partitions of R, the set of real numbers, and the correct reason for it: the negative real numbers, {0}, the positive real numbers

The given collection of sets forms a partition of R as these sets are pairwise disjoint and their union is R.

Identify whether the following collection of subsets is a partition of S= {−3,−2,−1, 0, 1, 2, 3} and the correct reason for it {−3,−2,−1}, {0, 1, 2, 3}

The given collection of sets forms a partition of S as these sets are mutually disjoint and the union of these sets is S.

The first terms of a sequence are: 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, . . . The next three terms are:

The next three terms are 0, 0, 0.

Terms of a Sequence: The first three terms of a sequence are: 2, 4, 16,

The next three terms are 256, 65536, 4294967296.

The range of the function that assigns to a bit string the number of ones minus the number of zeros is:

The set of all integers.

Identify the correct statements. Given the following graph the domain of the function f is

The set of all real numbers

Let A be the set of students who live within one mile of school and let B be the set of students who walk to classes. Describe the students in the set B -

The set of students who live more than a mile from school but nevertheless walk to class

Let A be the set of students who live within one mile of school and let B be the set of students who walk to classes. Describe the students in the set B - A

The set of students who live more than a mile from school but nevertheless walk to class

Let A be a 3 × 4 matrix, B be a 4 × 5 matrix, and C be a 4 × 4 matrix. Determine the product and the size of CB

This is defined, and its dimension is 4 × 5.

Decide whether the given relation on the set {1, 2, 3, 4}, is reflexive, symmetric, antisymmetric, or transitive. {(2, 2), (2, 3), (2, 4), (3, 2), (3, 3), (3, 4)}

Transitive

Consider the relation {(a, b) | aand b have the same age} on the set of all people. The relation R is an equivalence relation

True

if A, B, and Care sets such that A ∪ C = B ∪ C and A ∩ C = B ∩ C, we can conclude that A = B.

True

Identify the Venn diagram that illustrate the relationship between A ⊆B and B ⊆ C.

_____________________ | | | a b | |_______________c__U|

A person deposits $1,000 in an account that yields 9% interest compounded annually. How much money will the account contain after 100 years?

a100≈ $5,529,041

Find r, a1 and an expression for the nth term of the geometric sequence where a4 = -240 and a7 = 1920.

a1=30, r=-2, an=-15(-2)n

Find the twenty-third term of the arithmetic sequence -8, -3, 2, 7, .

a23=107

Find the sixth term of a geometric sequence where a1 = 10 and a2 = 12.

a6 = 12(6/5)4

A factory makes custom sports cars at an increasing rate. In the first month only one car is made, in the second month two cars are made, and so on with n cars made in the nth month. Set up a recurrence relation for the number of cars produced in the first n months by the factory..

an = an-1 + 1 ,a0 = 0

A person deposits $1,000 in an account that yields 9% interest compounded annually. Set up a recurrence relation for the amount in the account at the end of n

an= 1.09an - 1

Write an explicit formula for the pattern 19,22,25,28,31,34

an= 19+3n

The sequence {an} is a solution of the recurrence relation an= 8an − 1 − 16an − 2 if

an= 4n

The 5th term of an arithmetic sequence is 10, and the 14th term is 46. Give a recursive formula for the sequence.

an= an-1 +4

The 8th term of an arithmetic sequence is 8, and the 20th term is 44. Find the first term a0 and the common difference d.

an=-16, d=3

Write a formula for the nth term of the geometric sequence 5, 4 16/5, 64/25, ...

an=5(4/5)n

A factory makes custom sports cars at an increasing rate. In the first month only one car is made, in the second month two cars are made, and so on with n cars made in the nth month. Find an explicit formula for the number of cars produced the first n months by the factory.

an=n(n+1)/2

Determine which property is not satisfied by the relation R defined on the set of all real numbers where (x, y) ∈ R if and only if x − y is a rational number

antisymmetric

Given the following graph, identify the correct statements

f(0)=1

Identify the correct statements for the given functions from the set {a, b, c, d} to itself.

f(a) = d, f(b) = b, f(c) =c, f(d) = d is not a one-to-one function, as d is an image of more than one element.

Which of the following functions from Z to Z are one-to-one.

f(n) = ⌈n2⌉

From the given functions that are mapped from Z to Z, identify the onto functions.

f(n) =⌈n/2⌉

Which of these functions from R to R are bijections?

f(x) = x5+ 1

Let S be a subset of a universal set U. The characteristic function fS of S is the function from U to the set {0, 1} such that fS(x) = 1 if x belongs to S and fs(x) = 0 if x does not belong to S. Let A and B be sets. For all x ∈ U,

fA∩B(x) = fA(x) · fB(x)

Let f and g be functions from the set of real numbers to the set of real numbers. We say that f(x) is O(g(x)) (This is read "f(x) is big-oh of g(x)") if there are constants C and k such that |f(x)|≤ C|g(x)| whenever x>k. n^2 is O (n)

false

if A, B, and Care sets such that A ∪ C = B ∪ C, we can conclude that A = B.

false

How many different elements does A× B have if A has m elements and B has n elements?

mn

Decide whether the given relation on the set {1, 2, 3, 4}, is reflexive, symmetric, antisymmetric, or transitive {(1, 3), (1, 4), (2, 3), (2, 4), (3, 1), (3, 4)

none of the above

Matrices can be used to represent graphs. Consider a square matrix with dimension n, where n is the number of vertices of the graph. If the vertex i is connected with the vertex j, the entry cij=1 and =0 otherwise. Use this kind of matrix to describe the following graph:

pic 3rd one, 01000110 00111001 0101000 00110100 00101010 01001010 010001000 11000000

Determine whether the relation Ron the set of all real numbers is reflexive, symmetric, antisymmetric, and/or transitive, where (x, y) ∈ R if and only if x + y = 0.

symmetric

Let f and g be functions from the set of real numbers to the set of real numbers. We say that f(x) is O(g(x)) (This is read "f(x) is big-oh of g(x)") if there are constants C and k such that |f(x)|≤ C|g(x)| whenever x>k. 7x^2 is O (x^3)

true

Let f and g be functions from the set of real numbers to the set of real numbers. We say that f(x) is O(g(x)) (This is read "f(x) is big-oh of g(x)") if there are constants C and k such that |f(x)|≤ C|g(x)| whenever x>k. Determine the truth value of the statement: f(x)=x2+2x+1 is O(x2).

true

Which of the following defines an equivalence relation in the set of all people.

x is related to y if they have the same birthday.

Find the power set of the set {Ø, {Ø}}

{ Ø, {Ø}, { {Ø} }, {Ø, {Ø} } }