Math 243 Practice test 2

If f is the absolute value function f: ℝ → ℝ; f(x) = |x|, determine the pre-image f⁻¹([-2,1)).

(-1,1)

Simplify [-1, 3] ∪ (-2, 0) ∪ [1, 4).

(-2, 4)

Solve the inequality 1 < ⌈ 2x+1⌉ < 6.

(0,2]

Consider the statements 1. ∅ ∈ ∅ 2. ∅ ⊆ ∅.

1. is false, 2. is true.

Consider the statements: 1. Every set is a subset of itself. 2. Every set is a proper subset of itself.

1. is true, 2. is false.

How many elements does the set {0,{1,{2,3}}} contain?

2 elements

If A has 5 elements, how many elements does the power set of A have?

32

If A and B are sets, |A| = 10 and |B| = 5, then |A × B| = ?

50 (with margin: 0)

Check all correct ways of evaluating

A

Given a positive integer n, evaluate

A

If A and B are sets, simplify A ∩ (A ∪ B).

A

What do you get when you index shift the sigma sum ∑ k = 5 9 ( k + 7 ) 3 so that k starts at 0?

A

Given A = {1,2} and B = {1,3}, find A × B.

A × B = { (1,1), (1,3), (2,1), (2,3) }

Suppose A and B are sets. Match sets that must be equal, based on the given information.

A, B, A and B, A or B A ∪ ∅ ∪ ( B ∩ ∅ ) = A ∪ ( B ∩ ∅ ) because a union with the empty set has no effect. A ∪ ( B ∩ ∅ ) = A ∪ ( ∅ ) because an intersection with the empty set is always empty. A ∪ ( ∅ ) = A because a union with the empty set has no effect. To simplify B - ( A - B ), we remember that the set difference A - B can be expressed as A ∩ Bᶜ (complement of B). Therefore, B - ( A - B ) = B ∩ (A -B)ᶜ = B ∩ ( A ∩ Bᶜ )ᶜ. By De Morgan, ( A ∩ Bᶜ )ᶜ = Aᶜ ∪ B. Therefore, B - ( A - B ) = B ∩ (Aᶜ ∪ B). By the absorption law, B ∩ (Aᶜ ∪ B) = B. ( A ∩ B ) ∪ ( A ∩ B ) = A ∩ B by the idempotent law. To simplify ( A - B ) ∪ ( A ∩ B ) ∪ ( B - A ), we again express the set difference using the complement: A - B = A ∩ BᶜB - A = Aᶜ ∩ B. Now we reverse distribute and use the identity law: ( A - B ) ∪ ( A ∩ B ) = (A ∩ Bᶜ) ∪ ( A ∩ B ) = A ∩ (Bᶜ ∪ B) = A ∩ U = A. Therefore, using the associative law, ( A - B ) ∪ ( A ∩ B ) ∪ ( B - A ) = A ∪ ( B - A ) = A ∪ ( Aᶜ ∩ B ). Finally, we use the distributive law, followed by the identity law: A ∪ ( Aᶜ ∩ B ) = (A ∪ Aᶜ) ∩ (A ∪ B ) = U ∩ (A ∪ B ) = A ∪ B. Question 41.75 / 1.75 pts The two sets {1, 2} and {2, 1} are equal. Correct! True False The set is an unordered data structure. Both notations represent the set that contains the two numbers 1 and 2.

Which methods may we use for correctly evaluating the sum 3⁴ + 3⁵ + .. + 3⁹ ?

A,B,.C

Given the formula f(x) = x², pick all domain/codomain pairs A,B that would make f: A → B bijective.

All is correct

Identify all mistakes made in the following proof that for each integer n, there is an integer k such that n < k < n+2. Suppose n is an arbitrary integer. Therefore, k = n + 1 for all integers n. This means that n < n + 1 < n + 2. This proves that an integer k exists.

All is correct excepth there is nothing wrong with this proof

Pick all that apply. Any function f: {0} → {0} is..

All of them

Check the properties that the function f: [-2, 0) → [0,4], f(x) = x² has.

Decreasing,Strickly decreasing,injective

Let f:R→R;f(x)=x². Evaluate the two sets f((-1,1)) and f⁻¹((0,1]).

E, and something else

Is the following a correct proof that for every integer x, x + y = 2 has an integer solution y? Suppose x = 1. Then x + y = 2 has y = 1 as a solution. Select True for yes, False for no.

False

Is the following a correct proof that there is an integer solution of 3n + 5 = 8? Suppose 3n+5=8 for some integer n. Then 3n=3 and n=1. Select True for yes, False for no.

False

True or False? If a set is empty, so is its power set.

False

True or false? If A and B are sets, then A × B = B × A.

False

True or false? If a and b are real numbers and ⌊ ⌋ represents the floor function, then ⌊ab⌋ = ⌊a⌋ ⌊b⌋.

False

True or false? If a set A has 5 elements, and a set B has 7 elements, then the union A∪B must have 5+7=12 elements.

False

True or false? {∅} is the empty set.

False

The interval [1,2] is equal to the statement 1 ≤ x ≤ 2.

False It is true that the statement defines the membership condition for the set, i.e. it defines whether a number x is in the set. Thus, [1,2] = {x | 1 ≤ x ≤ 2 }.

How many elements are in the intersection of (2,4) and (3,5) ?

Infinitely many

Consider the sequence aₙ = 3·11ⁿ.

Is is geometric with common quotient 11.

Consider the sequence aₙ = 2+5n.

It is arithmetic with common difference 5.

Write a short proof up to three sentences for the statement ∃x∀y(x+y = y). The domain of discourse is the real numbers. First write your proof on a piece of scratch paper, then assemble the proof below. First sentence: Suppose y is an arbitrary real numbers. Second sentence: Pick x=0. Third sentence: Then x+y=y.

Let x be 0, assume y is an arbitrary real number, then x +y =y

Determine which one of the following students answered the following problem correctly: Is the function f: [0, 1] → ℝ; f(x) = x² injective? Is it surjective? Prove both of your answers based on the definitions of injective and surjective.

Sun

Check all that apply.

The rational numbers are a subset of the real numbers. The rational numbers are a proper subset of the real numbers.

Select all sets that are complement pairs (i.e. the two sets are complements of each other). The universal set U is given in each situation.

The set of rational numbers, the set of irrational numbers (U = the set of all real numbers). The set of even integers, the set of odd integers (U = the set of all integers).

Consider the statements 1. ∅ ∈ 𝒫 (∅) 2. ∅ ⊆ 𝒫 (∅). The notation 𝒫 (S) means the power set of S.

They are both true. The power set of ∅ is {∅, {∅}}. ∅ is an element of that. ∅ is a subset of any set.

On a piece of scratch paper write a short proof for the following statement. After you are done with this practice test, you will find a reference solution to this proof in the detailed response feedback. If you are not certain whether the proof you developed was correct, post it on Discussions. For all positive integers n, there is an even integer k such that n - 1/n < k < n + 2 + 1/n. Answer True if you developed a solution on paper for this problem as instructed.

True

The two sets {1, 2} and {2, 1} are equal.

True

True or false? The sum of the squares of the first n positive integers is n(n+1)(2n+1)/6.

True

Check all true statements.

You can always modify a non-injective function f: A→B to become injective, by replacing A by a suitable proper subset of A. By redefining the codomain of a function to make it equal to its range, you can always force the function to become surjective.

Solve the inequality 1 ≤ ⌊2x+1⌋ ≤ 6.

[0, 3)

If f: [-2,2] → B; f(x) = x² is surjective, then B =

[0,4]

Pick all that apply. A constant function is always..

increasing,decreasing

The function f: [0, ∞) → ℝ; f(x) = x² + 1 is

injective but not surjective

The function f: ℝ→ ℝ ; f(x) = x² + 1 is

neither surjective nor injective

Write a short proof up to three sentences for the statement ∀x∃y(y < 2x+1) and y is odd. In your proof you can use without proof simple theorems such as "even+even is even", "even + odd is odd" etc. The domain of discourse is integers. First write your proof on a piece of scratch paper, then assemble the proof below. First sentence: Suppose x is an arbitrary integer. Second sentence: Select y=2x. Third sentence: (The proof is complete.)

suppose x is an arbitrary interger select y = 2x-1 then y is odd and y = 2x-1 < 2x+1

If we visualize ℝ² as the plane, then ℤ² is

the grid of points (x,y) with integer coordinates. ℤ² means ℤ×ℤ, which by definition of Cartesian product is all points (x,y) with integers x and y. That set of points forms a grid in the plane.

We say that two sets are disjoint iff..

their intersection is the empty set.

If the universal set is [0,2], what is the complement of (0,1)?

{0} ∪ [1,2]

If f is the ceiling function from ℝ to ℝ, what is f((1/2, 3/2))?

{1,2}

What is the power set of the power set of {1} ?

{∅, {∅}, {{1}}, {∅, {1}}} The power set of {1} is {∅,{1}}. If we let A = ∅ and B = {1}, then the power set of {1} is {A,B}. We find the power set of that to be { ∅, {A}, {B}, {A,B} }. Substituting the definitions of A and B, we find the power set of the power set of {1} to be { ∅, {∅}, {{1}}, {∅, {1}}}.

Simplify {∅} ∪ ∅.

{∅}

Use a summation formula we learned in class to compute the sum of the integers from 1000 to 5000. Write the answer in un-simplified form.

½·5000·5001 - ½·999·1000

Given a universal set U, the complement of U is..

∅

Suppose A is a set. Simplify A × ∅.

∅