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Sets

Elements, subsets and set equality, worked through finite and infinite examples before the reserved symbols for the naturals, integers, rationals and reals.

A Complete Introduction to Abstract Algebra, Sets

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7 exercises

Exercise 1

Let SS be a non-empty set. Determine whether the following statements are true or false.

  1. {1,2}⊂{1,2,3,4}\{1,2\}\subset\{1,2,3,4\}

  2. {1,2,3}⊆{1,2,4}\{1,2,3\}\subseteq\{1,2,4\}

  3. ∅⊆∅\varnothing\subseteq\varnothing

  4. S⊂SS\subset S

  5. S∈{S,∅}S\in\{S,\varnothing\}

  6. {S}∉{S,∅}\{S\}\notin\{S,\varnothing\}

  7. S⊆{S,∅}S\subseteq\{S,\varnothing\} where S≠{∅}S\neq\{\varnothing\}

  8. {S}⊆{S,∅}\{S\}\subseteq\{S,\varnothing\}

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Exercise 2

Let S={1,2,3,4}S=\{1,2,3,4\}, T={2,3,5}T=\{2,3,5\}, U={(2,2),(3,3),(5,5)}U=\{(2,2),(3,3),(5,5)\}. Determine whether the following statements are true or false.

  1. S∪T={1,2,3,4,5}S\cup T=\{1,2,3,4,5\}

  2. S∪U={1,2,3,(5,5)}S\cup U=\{1,2,3,(5,5)\}

  3. S∩T={2,3}S\cap T=\{2,3\}

  4. T∩U=∅T\cap U=\varnothing

  5. S∖T={1,4}S\setminus T=\{1,4\}

  6. S∖{1,4}=TS\setminus\{1,4\}=T

  7. T2=UT^{2}=U

  8. U⊂(S∪T)2U\subset(S\cup T)^{2}

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Exercise 3

Let the sets

S\displaystyle S ={x∈Q∣x∈(−∞,0]} and\displaystyle=\{x\in\mathbb{Q}\mid x\in(-\infty,0]\}\text{ and} T\displaystyle T ={y∈Z∣y∈[−2,10] and y is an even number}.\displaystyle=\{y\in\mathbb{Z}\mid y\in[-2,10]\text{ and }y\text{ is an even number}\}.

List the elements in the set S∩TS\cap T.

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Problem 1

Let AA and BB be finite sets with cardinality nn. For what value(s) of nn can we be sure that A=BA=B?

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Problem 2

Let the sets

A\displaystyle A ={x∈R∣x2−x−2≤0} and\displaystyle=\{x\in\mathbb{R}\mid x^{2}-x-2\leq 0\}\text{ and} B\displaystyle B ={x∈[0,∞)∣12−x−x2>0}.\displaystyle=\{x\in[0,\infty)\mid 12-x-x^{2}>0\}.

  1. Express A∩BA\cap B in interval notation.

  2. Express A∪BA\cup B in interval notation.

  3. Express B∖AB\setminus A in interval notation.

  4. Is A∖B⊂[−1,0)A\setminus B\subset[-1,0)? Explain.

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Problem 3

Let the sets

A\displaystyle A ={(x,y)∈Z2∣5x+2y+3=0} and\displaystyle=\{(x,y)\in\mathbb{Z}^{2}\mid 5x+2y+3=0\}\text{ and} B\displaystyle B ={(x,y)∈[0,1]2∣2x2+5x+2y+1=0}.\displaystyle=\{(x,y)\in[0,1]^{2}\mid 2x^{2}+5x+2y+1=0\}.

Find the cardinality of the following sets. If the set is finite, list all elements of the set.

  1. A∩BA\cap B

  2. A∪BA\cup B

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Problem 4

Show that

for any sets AA, BB, and CC.

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Adapted from A Complete Introduction to Abstract Algebra © 2022–2026 Kan Onn Kit. Licensed under CC BY-SA 4.0. Source: A Complete Introduction to Abstract Algebra. Imported at revision a559e8b.