MATH Study 3

Find the value of given function. Match the given functions. 1. ⌈0.1⌉ 2. ⌊−0.1⌋ 3. ⌈⌊1/2⌋ + ⌈1/2⌉ + 1/2⌉ 4. ⌈2.99⌉ Explanation Ceiling function of 0.1 = ⌈0.1⌉ = smallest integer that is greater than or equal to 0.1 Ceiling function of 2.99 = ⌈2.99⌉ = smallest integer that is greater than or equal to 2.99 ⌈⌊1/2⌋ + ⌈1/2⌉ + 1/2⌉ =⌈0 + 1+ 1/2⌉=⌈3/2⌉ = Ceiling function of 1.5 = smallest integer that is greater than or equal to 1.5 Floor function of -0.1 = ⌊−0.1⌋ = largest integer that is less than or equal to -0.1

1. 1 2. -1 3. 2 4. 3

Translate each of these quantifications into English and determine their truth value. ∃x ∈ Z (x + 1 > x) Explanation Z represents set of integers. (x + 1 > x) explains that there is an integer such that the number obtained by adding 1 to it is greater than the integer.

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.

Click and drag the given steps to their corresponding step number to prove the given statement. A ∩ B ∩ C ⊆ A ∩ B. Explanation Suppose x ∈ A ∩ B ∩ C. Then x is in all three of these sets. In particular, it is in both A and B. Thus, x is in A ∩ B.

Suppose x ∈ A ∩ B ∩ C. Then x is in all three of these sets. In particular, it is in both A and B. Thus, x is in A ∩ B.

Determine whether each of these sets is the power set of a set, where a and b are distinct elements. {Ø, {a}, {Ø, a} } Explanation Ø, {a}, and {Ø, a} are the subsets. These are not valid subsets of a set.

This cannot be the power set of any set.

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

This is the power set of {a, b}.

Let S = {-1, 0, 2, 4, 7}. f(x) = 1 g(x) = 2x + 1 h(x) = ⌈x5⌉ k(x) = ⌊x2+ 13⌋ Click and drag to identify the sets on the right with the function images on the left. Explanation f(S) = {1} g(S) = {-1, 1, 5, 9, 15} h(S) = {0, 1, 2} k(S) = {0, 1, 5, 16}

f(S) = {1} g(S) = {-1, 1, 5, 9, 15} h(S) = {0, 1, 2} k(S) = {0, 1, 5, 16}

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}) Explanation Inverse is f−1({1})f−1({1}) = {-1, 1}

f−1({1})f−1({1}) = {-1, 1}

Match each of the given combinations of sets A, B, and C to its corresponding Venn diagram. 1. (A− B) ∪ (A− C) ∪ (B− C) 2. A∩ (B ∪ C) 3. A¯∩ B ¯∩ C¯

1. https://imgur.com/a/zkX7SRk 2. https://imgur.com/a/UaJQzzy 3. https://imgur.com/a/5g61U6K

Find the cardinality of the set A. A = {ø, {ø}, {ø, {ø} } } Explanation Cardinality is the number of elements in a set. The set {Ø, {Ø}, {Ø, {Ø} } } contains three elements Ø, {Ø}, and {Ø, {Ø} }.

3, as this set contains exactly three different elements, each of which is a set

Find two sets A and B such that A ∈ B and A ⊆ B. Explanation If A ∈ B and A ⊆ B then let A = ∅ and B = {∅} as the simplest example.

A = ∅ and B = {∅}

Determine whether each of these sets is finite, countably infinite, or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set. The integers that are multiples of 10 (Check all that apply.) Explanation The list of numbers of the set in the order are 0, 10, −10, 20, −20, 30, . . . .

Check the following: The set is countably infinite with one-to-one correspondence 1 ↔ 0, 2 ↔ 10, 3 ↔ −10, 4 ↔ 20, 5 ↔ −20, 6 ↔ 30, and so on.

Let A = {a, b, c}, B = {x, y}, and C = {0, 1}. Identify A × B × C. Explanation The ordered n-tuple (a1, a2, ..., an) is the ordered collection that has a1 as its first element, a2 as its second element, ..., and an as its nth element. The Cartesian product of the sets A, B, and C, denoted by A × B × C, is the set of all ordered 3-tuples (a1, a2, a3), where a1 ∈ A, a2 ∈ B, and a3 ∈ C. Hence, A × B × C = {(a1, a2, a3) ∣ a1 ∈ A ∧ a2 ∈ B ∧ a3 ∈ C}. Thus, the Cartesian product A × B × C can be written as {(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)}.

{(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 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 Explanation 0, 2, 4, 5, 6, 7, 8, 9, and 10 are the elemnts in the set (A ∩ B) ∪ C.

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

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. 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 Explanation 4, 5, 6, 8, and 10 are the elements in the set (A ∪ B) ∩ C.

{4, 5, 6, 8, 10}

Match the sets on the right column to the sets on the left column by clicking and dragging them. Explanation The notation{a, b, c, d}represents the set with the four elements a, b, c, and d. This way of describing a set is known as the roster method. The given sets can be arranged in the roster method as follows: {x | x is a real number such that x2 = 1} = {1, -1} {x | x is a positive integer less than 12} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} {x | x is the square of an integer and x < 100} = {0, 1, 4, 9, 16, 25, 49, 64, 81} {x | x is an integer such that x2 = 2} = {∅}

{x | x is a real number such that x2 = 1} = {1, -1} {x | x is a positive integer less than 12} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} {x | x is the square of an integer and x < 100} = {0, 1, 4, 9, 16, 25, 49, 64, 81} {x | x is an integer such that x2 = 2} = {∅}