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Chapter practice · Introduction to Analysis | Math Infinitum

Intuition

Bring the chapter together: read membership statements, compare sets, and compute operations. These questions combine familiar ideas without a worked example immediately above the answer choices.

Think of checking a filter step by step: identify the starting objects, apply each condition, and verify the final result against the original request.

Before you practise

For each question, use the sets given in that question. Identify the universal set before taking a complement. For a compound expression, evaluate parentheses first, then apply the remaining operation.

$A\subseteq B$ : every element of A also belongs to B.

How to use the definitions

Union $A\cup B$ uses or; intersection $A\cap B$ uses and.
Difference $A\setminus B$ keeps elements of A that are absent from B.
Within a specified U, $A^c=U\setminus A$ .

Applications

Practice

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Let $A=\{0,2,2,5\}$ . How many distinct elements does A have?

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Let $A=\{1,4\}$ and $B=\{1,4,7\}$ . Which statement is true?

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Let $A=\{2,6\}$ and $B=\{6,8\}$ . Find $A\cup B$ .

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For $A=\{2,6\}$ and $B=\{6,8\}$ , find $A\cap B$ .

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Let $A=\{1,2,4\}$ and $B=\{2,3\}$ . Find $A\setminus B$ .

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Fix $U=\{0,1,2,3,4,5\}$ and let $A=\{1,3,5\}$ . Find $A^c$ within U.

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Fix $U=\{1,2,3,4,5,6\}$ , $A=\{1,2,3\}$ and $B=\{3,4\}$ . Find $(A\cup B)^c$ within U.

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Fix $U=\{1,2,3,4\}$ , $A=\{1,2\}$ and $B=\{2,3\}$ . Find $A\cap B^c$ .

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Let $A=\{1,2,3,4\}$ and $B=\{x,y\}$ . How many elements does $A\times B$ have?

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Let $f\colon\mathbb{Z}\to\mathbb{Z}$ , $f(n)=n^2$ , and $S=\{-2,-1,1,2\}$ . What is $f(S)$ ?

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For the same $f(n)=n^2$ on $\mathbb{Z}$ , what is $f^{-1}(\{1,2\})$ ?

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$f\colon\mathbb{R}\to\mathbb{R}$ , $f(x)=x^3$ , is injective.

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Which of these four sets cannot be written as a list?

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Let $A_n=\{n,n+1,n+2,\dots\}$ for $n\in\mathbb{N}$ . What is $\bigcap_{n=1}^{\infty}A_n$ ?

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Completion

Lesson complete

Great work! You now know how to:

Combine membership, inclusion and the set operations in one question
Compute an image and a preimage
Recognise which familiar sets are countable