Practice Finite vs Infinite in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

Finite describes a quantity or set with a definite end; infinite describes something that goes on forever without bound.

A jar of 100 marbles is finite. The counting numbers are infinite.

Showing a random 20 of 50 problems.

Example 1

medium
A sequence is defined by an=1na_n=\frac{1}{n} for n=1,2,3,…n=1,2,3,\ldots. Is the sequence finite or infinite, and is its set of values bounded?

Example 2

challenge
Show that the set of points (x,y)(x, y) with x,yx, y rational and x2+y2=1x^2 + y^2 = 1 is infinite.

Example 3

easy
Finite or infinite: the multiples of 55, i.e. 5,10,15,…5,10,15,\ldots?

Example 4

hard
Is the set of all finite-length strings using letters {a,b}\{a, b\} finite or infinite? Countable?

Example 5

medium
How many primes are there? Finite or infinite? Name the theorem.

Example 6

medium
Is the set of whole numbers from 00 to 1,000,0001{,}000{,}000 finite or infinite? How many elements?

Example 7

easy
Is the number of even numbers between 11 and 1010 finite or infinite?

Example 8

medium
True or false: every subset of a finite set is finite.

Example 9

medium
Is the set of points inside a square of side 11 finite or infinite?

Example 10

easy
Finite or infinite: the set of integers Z\mathbb{Z}?

Example 11

medium
Why might two infinite sets be the 'same size'? Give the integers and even integers as an example.

Example 12

medium
Consider S={n:n is a multiple of 6 and n≤100}S = \{n : n \text{ is a multiple of } 6 \text{ and } n \le 100\}. Is SS finite or infinite, and how many elements does it have?

Example 13

challenge
Argue that the reals in (0,1)(0,1) cannot be listed (sketch the diagonal idea).

Example 14

medium
Are all infinite sets the same size? Name two infinities of different sizes.

Example 15

medium
A repeating decimal 0.142857‾0.\overline{142857} has how many distinct decimal digits in its expansion, and is the digit string finite or infinite?

Example 16

easy
Finite or infinite: the set of perfect squares 1,4,9,16,…1, 4, 9, 16, \ldots?

Example 17

medium
Classify and explain: the set of fractions 1n\frac{1}{n} for n=1,2,3,…n=1,2,3,\ldots

Example 18

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Finite or infinite: the set of rational numbers Q\mathbb{Q}? Is it countable?

Example 19

hard
Is the set of all polynomials with integer coefficients finite or infinite? Countable?

Example 20

medium
Finite or infinite: the set of solutions to x2=9x^2 = 9 over the integers?
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