COT 3100 Homework 6

Translate each of these quantifications into English and determine their truth value. ∃x ∈ R (x^3 = −1)

Q(x): There is a real number whose cube is −1. Q(x) is true.

Translate each of these quantifications into English and determine their truth value. ∃x ∈ Z (x + 1 > x)

Q(x): There is an integer such that the number obtained by adding 1 to it is greater than the integer. Q(x) is true.

Let A = {a, b, c}, B = {x, y}, and C = {0, 1}. Identify A × B × C.

{(a, x, 0), (a, x, 1), (a, y, 0), (a, y, 1), (b, x, 0), (b, x, 1), (b, y, 0), (b, y, 1), (c, x, 0), (c, x, 1), (c, y, 0), (c, y, 1)}

Let Ai = {..., −2, −1, 0, 1, ..., i}. Identify n∩(i=1)Ai.

{..., -2, -1, 0, 1}

The successor of the set A is the set A ∪ {A}. Identify the successor of the given set. {1, 2, 3}

{1, 2, 3, {1, 2, 3}}

Find the symmetric difference of {1, 3, 5} and {1, 2, 3}.

{2, 5}

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 the value of given function. A ∩ B ∩ C

{4, 6}

From the given functions from Z × Z to Z, identify the onto functions. (Check all that apply.)

- f(m, n) = 2m - n - f(m, n) = m + n + 1 - f(m, n) = |m|− |n|

Find the value of given function. ⌈−0.1⌉

0

Find the value of given function. ⌊1.1⌋

1

Consider the sets A = {a, b, c, d, e}, B = {b, c, d, g, p, t, v}, C = {c, e, i, o, u, x, y, z}, and D = {d, e, h, i, n, o, t, u, x, y}. Each of the given sets can be represented using a bit string of length 26, where ith bit represents the ith letter of the alphabet. Match each set on the left to the corresponding bit string on the right.

A = 11 1110 0000 0000 0000 0000 0000 B = 01 1100 1000 0000 0100 0101 0000 C = 00 1010 0010 0000 1000 0010 0111 D = 00 0110 0110 0001 1000 0110 0110

Identify A^2 for the given set A. A = {0, 1, 3}

{(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (1, 3), (3, 0), (3, 1), (3, 3)}

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 each of these sets. B - A

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

Identify ∞∪(i=1)Ai and ∞∩(i=1)Ai for the given Ai, where i is a positive integer. Ai = {-i, i}

The union is Z - {0} and the intersection is Ø.

Find the power set of each of the following sets, where a and b are distinct elements. {b}

{ Ø, {b} }

Determine whether each of these sets is the power set of a set, where a and b are distinct elements. Ø

This cannot be the power set of any set.

Let f be a function from the set A to the set B. Let S be a subset of B. We define the inverse image of S to be the subset of A whose elements are precisely all pre-images of all elements of S. We denote the inverse image of S by f−1f−1 (S), so f −1(S) = {a ∈ A ∣ f(a) ∈ S}. Let f be the function from R to R defined by f(x) = x2. Find the value of given functions. f^−1({1})

f^−1({1}) = {-1, 1}

Let f be a function from the set A to the set B. Let S be a subset of B. We define the inverse image of S to be the subset of A whose elements are precisely all pre-images of all elements of S. We denote the inverse image of S by f−1f−1 (S), so f −1(S) = {a ∈ A ∣ f(a) ∈ S}. Let f be the function from R to R defined by f(x) = x2. Find the value of given functions. f^−1(x| 0< x< 1)

f−1(x| 0< x< 1)f−1(x| 0⁢< x⁢< 1) = {x| -1 < x < 0 ∨ 0 < x < 1}

Which of these statements is correct about the function g if both of the functions f and f∘g are one-to-one?

g is one-to-one as f ο g is one-to-one.

Consider the sets A = {a, b, c, d, e}, B = {b, c, d, g, p, t, v}, C = {c, e, i, o, u, x, y, z}, and D = {d, e, h, i, n, o, t, u, x, y}. Identify how bitwise operations on bit strings can be used to find the combination (A ∪ D) ∩ (B ∪ C).

(11 1110 0000 0000 0000 0000 0000 ∨ 00 0110 0110 0001 1000 0110 0110) ∧ (01 1100 1000 0000 0100 0101 0000 ∨ 00 1010 0010 0000 1000 0010 0111)

Let f be a function from A to B and S be a subset of B. Identify the correct steps involved in showing that f⁻¹(S̅) = f̅⁻¹(̅S̅)̅.

- We know that S' = B−S and (f^−1(S))' = A − f^−1(S). - Therefore, by the definition of inverse image, x ∉ f^−1(S); so, (x ∈ f^−1(S))'. - Let x∈f−1(S'). This means that f(x)∈S' or equivalently that f(x)∉S. - Assume that (x ∈ f^−1(S))'. Then, x ∉ f^−1(S). This means that f(x) ∉ S. - It shows that f(x)∈S' . Therefore, by the definition of inverse image, x∈f−1(S').

Let f be a function from A to B. Let S and T be the subsets of B. Identify the correct steps involved in showing that f^−1(S ∩ T) = f^−1(S) ∩ f^−1(T). (Check all that apply.)

- We need to prove two things. First, suppose that x ∈ f^−1(S ∩ T). This means that f(x) ∈ S ∩ T. - Therefore, f(x) ∈ S and f(x) ∈ T. Thus, x ∈ f^−1(S) and x ∈ f^−1(T). - Then, x ∈ f^−1(S) ∩ f^−1(T). Thus, we have shown that f^−1(S ∩ T) ⊆ f^−1(S) ∩ f^−1(T). - Conversely, suppose that x ∈ f^−1(S) ∩ f^−1(T). Then, x ∈ f^−1(S) and x ∈ f^−1(T); so, f(x) ∈ S and f(x) ∈ T. - Thus, we know that f(x) ∈ S ∩ T; so, by definition, x ∈ f^−1(S ∩ T). This shows that f^−1(S) ∩ f^−1(T) ⊆ f^−1(S ∩ T), as desired.

Find the truth set of each of the given predicates if the domain is a set of integers. P(x): x^3 ≥ 1

{x ∈ Z | x ≥ 1} {1, 2, 3, 4, 5, 6, ...}

Use set builder notation to give a description of each of these sets. {−3, −2, −1, 0, 1, 2, 3}

{x ∈ Z | −3 ≤ x ≤ 3}

The successor of the set A is the set A ∪ {A}. Identify the successor of the given set. Ø

{Ø}

Let f be a function from A to B. Let S and T be the subsets of B. Identify the correct steps involved in showing that f^−1(S ∪ T) = f^−1(S) ∪ f^−1(T). (Check all that apply.)

- We need to prove two things. First, suppose that x ∈ f^−1(S ∪ T). This means that f(x) ∈ S ∪ T. - Therefore, either f(x) ∈ S or f(x) ∈ T. Then, x ∈ f^−1(S) or x ∈ f^−1(T). - In either case, x ∈ f^−1(S) ∪ f^−1(T). Thus, we have shown that f^-1(S ∪ T) ⊆ f^−1(S) ∪ f^−1(T). - Conversely, suppose that x ∈ f^−1(S) ∪ f^−1(T). Then, either x ∈ f^−1(S) or x ∈ f^−1(T); so, either f(x) ∈ S or f(x) ∈ T. - Thus, we know that f(x) ∈ S ∪ T; so, by definition, x ∈ f^−1(S ∪ T). This shows that f−1(S) ∪ f^−1(T) ⊆ f^−1(S ∪ T), as desired.

Identify the correct statements for the given functions from the set {a, b, c, d} to itself. (Check all that apply.)

- f(a) = b, f(b) = a, f(c) = c, f(d) = d is a one-to-one function, as each element is an image of exactly one element. - f(a) = b, f(b) = b, f(c) = d, f(d) = c is not a one-to-one function, as b is an image of more than one element. - 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 these functions from R to R are bijections? (Check all that apply.)

- f(x) = -3x + 4 - f(x) = x^5 + 1

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, then match the unknowns—A, B, C, and D—on the right to their corresponding values on the left. 1) A 2) D 3) C 4) B

1) 4 2) 3 3) 0 4) 5

Suppose that A is the set of sophomores at your school, B is the set of students in discrete mathematics at your school, and the universal set U is the set of all students at your school. Match the sets given in the left to their symbolic expression in the right. 1) The set of sophomores taking discrete mathematics in your school 2) The set of students at your school who either are sophomores or are taking discrete mathematics 3) The set of sophomores at your school who are not taking discrete mathematics 4) The set of students at your school who either are not sophomores or are not taking discrete mathematics

1) A ∩ B 2) A∪B 3) A - B 4) A' ∪ B'

Click and drag the given steps to their corresponding step number to prove the given statement. A ∪ B ⊆ A ∪ B ∪ C.

1) Assume x ∈ A ∪ B. 2) Then x ∈ A or x ∈ B by definition of union. 3) Therefore, x is in A or in B or in C by the addition rule. 4) Therefore, x is in A ∪ B ∪ C by definition of union.

Prove De Morgan's law by showing that (A∪ B)' = A' ∩ B' if A and B are sets. Prove De Morgan's law by showing that each side is a subset of the other side by considering x∈(A∪B)' .

1) Assume x∈(A∪B)' ⇔ x∉A∪B 2) Hence, −(x∈A∨x∈B) is true. 3) Applying De Morgan's law of proposition, we get -(x∈A) ∧ −(x∈B). 4) Then, we can write x∉A ∧ x∉B. 5) Using the definition of the complement of a set, x∈A' ∧ x∈B'. 6) Using the definition of intersection x∈ A' ∩ B'. 7) Therefore, we get x∈(A∪ B)' → x∈ A' ∩ B' .

Click and drag the domain and range on the left to their corresponding functions defined on the right, provided lambda (λ) is the empty string. 1) The function that assigns to each pair of positive integers the first integer of the pair 2) The function that assigns to each positive integer its largest decimal digit 3) The function that assigns to a bit string the number of ones minus the number of zeros in the string 4) The function that assigns to each positive integer the largest integer not exceeding the square root of the integer 5) The function that assigns to a bit string the longest string of ones in the string

1) Domain: Z+ * Z+ and range: Z+ 2) Domain: Z+ and range: {1, 2, 3, 4, 5, 6, 7, 8, 9} 3) Domain: set of all bit strings, and range: Z 4) Domain: Z+ and range: Z+ 5) Domain: set of bit strings, and range: {λ, 1, 11, 111, ...}

Click and drag the given steps to their corresponding step number to prove the given statement. A ∩ Ø = Ø

1) {x | x ∈ A ∧ x ∈ ∅} 2) {x | x ∈ A ∨ F} 3) {x | F} 4) Ø

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 each of these sets. A ∪ B

The set of students who either live within one mile of school or walk to class

Let Ai = {1, 2, 3, ..., i} for i = 1, 2, 3, .... Identify n∪(i=1)Ai.

A(n)

Prove De Morgan's law by showing that (A∪B) = A' ∩ B' if A and B are sets. Identify the the unknowns X, Y, Z, P, Q, and R in the given membership table.

X = 0 Y = 1 Z = 1 P = 1 Q = 0 R = 1