Virtual Labs

What property makes a set countable?

a: no decimal values Explanation

Explanation

b: no negative values Explanation

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c: bijection with natural numbers Explanation

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d: bijection with real numbers Explanation

Explanation

The set of all real numbers is ______.

a: infinite and countable Explanation

Explanation

b: finite and countable Explanation

Explanation

c: infinite and uncountable Explanation

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d: infinite and countable Explanation

Explanation

Which of these is an uncountable set?

a: integers Explanation

Explanation

b: rational numbers Explanation

Explanation

c: binary numbers Explanation

Explanation

d: none of the above Explanation

Explanation

Which of these is a countable set?

a: integers Explanation

Explanation

b: rational numbers Explanation

Explanation

c: binary numbers Explanation

Explanation

d: all of the above Explanation

Explanation

There are languages that can not be accepted by a turing machine

a: true Explanation

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b: false Explanation

Explanation

Given a set

S = \{1, 2, 3, 4\}

and an equivalence relation

R

S

, if we know that

[1] = \{1, 2\}

and

[3] = \{3, 4\}

, what can we conclude?

(2,1) \in R

but

(1,2) \notin R

Explanation

(2,3) \in R

and

(3,4) \in R

Explanation

(2,2) \notin R

but

(1,2) \in R

Explanation

(2,2) \in R

and

(1,2) \in R

Explanation

Let

R

be an equivalence relation on

\mathbb{Z}

defined by

aRb

if and only if

|a - b| \leq 2

. What is the size of the equivalence class

[0]_R

a: 2 Explanation

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b: 3 Explanation

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c: 4 Explanation

Explanation

d: 5 Explanation

Explanation

Let

R

be an equivalence relation on a finite set

A

with

|A| = 8

. If

R

has exactly three equivalence classes and one class contains 4 elements, what is the sum of all possible sizes for the second largest equivalence class?

a: 6 Explanation

Explanation

b: 7 Explanation

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c: 8 Explanation

Explanation

d: 9 Explanation

Explanation

Consider Cantor's diagonalization proof applied to the set of all infinite binary strings. If we have an enumeration

s_1, s_2, s_3, \ldots

and construct the diagonal string

d

where

d_i

is the opposite of the

i

-th bit of

s_i

, which statement is most accurate?

a: The diagonal string

d

must appear somewhere in the enumeration Explanation

Explanation

b: The diagonal string

d

differs from every string in the enumeration in at least one position Explanation

Explanation

c: The diagonal string

d

can be made to equal some string in the enumeration by careful construction Explanation

Explanation

d: The diagonal construction only works for finite sets Explanation

Explanation

Diagonalization