Practice Set in Math

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

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Quick Recap

A well-defined collection of distinct, unordered objects called elements, described either by listing or by a membership rule.

Think of a set as a bag that can hold anything — numbers, names, shapes — but with two strict rules: no duplicates allowed and the order in which items sit inside the bag does not matter.

Showing a random 20 of 50 problems.

Example 1

medium
Write {x∈Z:x divides 6}\{x \in \mathbb{Z} : x \text{ divides } 6\} in roster form (positive divisors only).

Example 2

medium
Set AA is described as 'all integers xx with x2=9x^2 = 9'. List AA by roster.

Example 3

medium
Write {x∈Z:x2=9}\{x \in \mathbb{Z} : x^2 = 9\} in roster form.

Example 4

easy
Is {3}\{3\} the same thing as the number 33?

Example 5

hard
How many subsets does a set with 55 elements have?

Example 6

easy
True or false: the set of letters in 'EYE' has 33 elements.

Example 7

challenge
Prove that if sets AA and BB satisfy A⊆BA \subseteq B and B⊆AB \subseteq A, then A=BA = B.

Example 8

easy
Is {1,2,3}\{1, 2, 3\} the same set as {3,2,1}\{3, 2, 1\}?

Example 9

challenge
Set SS contains its own number of elements rule: S={1,2,…,n}S = \{1, 2, \dots, n\} has exactly nn elements. If a set TT satisfies ∣T∣=∣T∪{5}∣|T| = |T \cup \{5\}|, what must be true about 55?

Example 10

medium
List the subset of {1,2,3,4,5,6,7,8,9,10}\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} consisting of perfect squares.

Example 11

easy
Let C={a,e,i,o,u}C = \{a, e, i, o, u\}. List all elements of CC and state the cardinality ∣C∣|C|.

Example 12

easy
Let A={2,4,6,8,10}A = \{2, 4, 6, 8, 10\}. Determine whether 6∈A6 \in A and whether 5∈A5 \in A.

Example 13

medium
Write the set B={x∈Z:−2≤x<3}B = \{x \in \mathbb{Z} : -2 \le x < 3\} in roster notation.

Example 14

medium
Convert {x∈Z:1≤x≤6 and x is odd}\{x \in \mathbb{Z} : 1 \le x \le 6 \text{ and } x \text{ is odd}\} to roster form.

Example 15

medium
True or false: {1,2,3}\{1, 2, 3\} has the same elements as {2,3,1,2}\{2, 3, 1, 2\}.

Example 16

easy
Is {1,2}={1,2,3}\{1, 2\} = \{1, 2, 3\}?

Example 17

easy
Are the sets {2,4,4,6}\{2, 4, 4, 6\} and {6,2,4}\{6, 2, 4\} equal? Explain.

Example 18

easy
Let A={0,1,2,3}A = \{0, 1, 2, 3\}. Is 0∈A0 \in A? Is −1∈A-1 \in A?

Example 19

medium
Are the sets {x:x is even}\{x : x \text{ is even}\} and {2k:k∈Z}\{2k : k \in \mathbb{Z}\} equal?

Example 20

medium
Is the collection of 'the three best movies ever' a well-defined set?
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