Calculus

Set Theory

Set Theory Logo

Introduction to Sets

The Concept of a Set

The basic concepts of set theory were created and developed in the late \(19\text{th}\) century by German mathematician Georg Cantor \(\left( {1845 – 1918} \right).\)

German mathematician Georg Cantor
Fig.1 Georg Cantor (1845-1918)

According to Cantor’s definition, a set is any collection of well defined objects, called the elements or members of the set.

Sets are usually denoted by capital letters \(\left( {A,B,X,Y, \ldots } \right).\) The elements of the set are denoted by small letters \(\left( {a,b,x,y, \ldots } \right).\)

If \(X\) is a set and \(x\) is an element of \(X,\) we write \(x \in X.\) The symbol \(\in\) was introduced by Italian mathematician Giuseppe Peano \(\left( {1858 – 1932} \right)\) and is an abbreviation of the Greek word \(\epsilon\sigma\tau\iota\) – “be”.

Italian mathematician Giuseppe Peano
Fig.2 Giuseppe Peano (1858-1932)

If \(y\) is not an element of \(X,\) we write \(y \notin X.\)

Defining Sets

There are two basic ways of describing sets – the roster method and set builder notation.

Roster Method

In roster notation, we just list the elements of a set. The elements are separated by commas and enclosed in curly braces. For example, \(X = \left\{ {1,5,17,286} \right\}\) or the set of natural numbers including zero \({\mathbb{N}_0} = \left\{ {0,1,2,3,4, \ldots } \right\}.\)

In a listing of the elements of a set, each distinct element is listed only once. The order in which elements are listed does not matter. For example, the following sets are equal:

\[\left\{ {5,5,6,7} \right\}\;and\;\left\{ {5,6,7} \right\};\]

\[{\left\{ {1,2,3} \right\},\left\{ {2,3,1} \right\}\;and\;}\kern0pt{\left\{ {3,2,1} \right\}}.\]

When writing infinite sets and there is a clear pattern to the elements, an ellipsis (three dots) can be used.

Set Builder Notation

In set builder notation, we define a set by describing the properties of its elements instead of listing them. This method is especially useful when describing infinite sets.

The notation includes on or more set variables and a rule defining which elements belong to the set and which are not. The rule is often expressed in the form of a predicate. The set variable and rule are separated by a colon “:” or vertical slash “|”.

Examples:

  1. The set of all uppercase letters of the English alphabet.
  2. \[{U = \left\{ {x\,|\,x \text{ is an uppercase letter}}\right.}\kern0pt{\left.{{\text{of the English alphabet}}} \right\}}\]
  3. The set of all prime numbers \(p\) less than \(1000.\)
  4. \[{P = \left\{ p\,|\,p \text{ is a prime number and}\right.}\kern0pt{\left. p \lt 1000 \right\}}\]
  5. The set of all \(x\) such that \(x\) is a negative real number.
  6. \[{X = \left\{ {x\,|\,x \in \mathbb{R} \text{ and } x \lt 0} \right\} \text{ or }}\kern0pt{ X = \left\{ {x \in \mathbb{R}\,|\,x \lt 0} \right\}}\]
  7. The set of all internal points \(\left( {x,y} \right)\) lying within the circle of radius \(1\) centered at the origin.
  8. \[C = \left\{ {x,y \in \mathbb{R}\,|\,{x^2} + {y^2} \lt 1} \right\}\]

Universal and Empty Sets

A set which contains all the elements under consideration is called the universal set and is denoted as \(U.\) The universal set is problem specific. For example, if a question is related to numbers, the universal set can be either all natural numbers \(\left( {U = \mathbb{N}} \right),\) or all integers \(\left( {U = \mathbb{Z}} \right),\) or all rational numbers \(\left( {U = \mathbb{Q}} \right),\) etc. – depending on the context.

There is a special name for the set which contains no elements. It is called the empty set and is denoted by the symbol \(\require{AMSsymbols}\varnothing.\) There is only one empty set.

Subsets

A set \(A\) is a subset of the set \(B\) if every element of \(A\) is also an element of the set \(B.\) A subset is denoted by the symbol \(\subseteq,\) so we write \(A \subseteq B.\) If a set \(C\) is not a subset of \(B,\) we write: \({C \not\subseteq B.}\)

The empty set \(\varnothing\) is a subset of every set, including itself.

The sets \(A\) and \(B\) are equal if simultaneously \(A \subseteq B\) and \(B \subseteq A.\) Notation: \(A = B\).

A set \(A\) is called a proper subset of \(B\) if \(A \subseteq B\) and \(A \ne B.\) In this case, we write \(A \subset B.\) Some authors use the symbol \(\subset\) to indicate any (proper and improper) subsets.

The power set of any set \(A\) is the set of all subsets of \(A,\) including the empty set and \(A\) itself. It is denoted by \(\mathcal{P}\left( A \right)\) or \({2^A}.\) If the set \(A\) contains \(n\) elements, then the power set \(\mathcal{P}\left( A \right)\) has \({2^n}\) elements.


Solved Problems

Click or tap a problem to see the solution.

Example 1

In every set, all elements, except one, have some property. Describe this property and find the element that does not have it.
  1. \({R = \left\{ {\text{red},\,\text{orange},\,\text{yellow},\,\text{brown},}\right.}\) \({\left.{\text{green},\text{blue},\text{indigo},\text{violet}} \right\}}\)
  2. \(S = \left\{ {2,5,10,13,17,26,37} \right\}\)
  3. \({P = \left\{ {\text{tetrahedron},\,\text{cube},\,\text{pyramid},}\right.}\) \({\left.{\text{octahedron},\,\text{dodecahedron},}\right.}\) \({\left.{\text{icosahedron}} \right\}}\)
  4. \(F = \left\{ {\large{\frac{2}{3}}\normalsize, \large{\frac{3}{8}}\normalsize, \large{\frac{4}{5}}\normalsize, \large{\frac{5}{{24}}}\normalsize, \large{\frac{6}{{35}}}\normalsize, \large{\frac{7}{{47}}}\normalsize} \right\}\)

Example 2

Write the following sets in roster form:
  1. \(A = \left\{ {n\,|\,n \in \mathbb{Z},\left| n \right| \lt 5} \right\}\)
  2. \(P = \left\{ {p\,|\,p \text{ is prime},\, p \lt 25} \right\}\)
  3. \(B = \left\{ {x \in \mathbb{N}\,|\,{x^2} + 4x – 5 = 0} \right\}\)
  4. \(C = \left\{ {x\,|\,x \in \mathbb{Z},\,{x^2} – 2x – 8 \le 0} \right\}\)
  5. \({D = \bigl\{ {\left( {x,y} \right)\,|\,x \in \mathbb{N}, y \in \mathbb{N},}\bigr.}\) \({\bigl.{{{\left( {x – 1} \right)}^2} + {y^2} \le 1} \bigr\}}\)

Example 3

How many elements in each of the sets:
  1. \(\varnothing\)
  2. \(\left\{\varnothing\right\}\)
  3. \(\left\{ {\varnothing,\left\{ \varnothing \right\}} \right\}\)
  4. \(\left\{ {\left\{ {\varnothing,\left\{ \varnothing \right\}} \right\}} \right\}\)
  5. \(\left\{ {\varnothing,\left\{ \varnothing \right\},\varnothing} \right\}\)

Example 4

Determine which of the statements are true and which are not:
  1. \(\varnothing \in \varnothing\)
  2. \(\varnothing \subseteq \varnothing\)
  3. \(\varnothing \in \left\{\varnothing\right\}\)
  4. \(\varnothing \subseteq \left\{\varnothing\right\}\)

Example 5

Find the power set of the following sets:
  1. \(A = \left\{ {1,\left\{ 1 \right\}} \right\}\)
  2. \(B = \left\{ {1,2,3,3} \right\}\)
  3. \(C = \left\{ {1,\left\{ {2,3} \right\},4} \right\}\)
  4. \(D = \left\{ {x \in \mathbb{Z}\,|\,{x^2} \lt 2} \right\}\)

Example 6

Let \(A \subseteq B\) and \(a \in A.\) Determine whether these statements are true or false:
  1. \(a \not\in B\)
  2. \(A \in B\)
  3. \(a \subseteq A\)
  4. \(\left\{a\right\} \subseteq A\)
  5. \(\left\{a\right\} \subseteq B\)

Example 1.

In every set, all elements, except one, have some property. Describe this property and find the element that does not have it.
  1. \({R = \left\{ {\text{red},\,\text{orange},\,\text{yellow},\,\text{brown},}\right.}\) \({\left.{\text{green},\text{blue},\text{indigo},\text{violet}} \right\}}\)
  2. \(S = \left\{ {2,5,10,13,17,26,37} \right\}\)
  3. \({P = \left\{ {\text{tetrahedron},\,\text{cube},\,\text{pyramid},}\right.}\) \({\left.{\text{octahedron},\,\text{dodecahedron},}\right.}\) \({\left.{\text{icosahedron}} \right\}}\)
  4. \(F = \left\{ {\large{\frac{2}{3}}\normalsize, \large{\frac{3}{8}}\normalsize, \large{\frac{4}{5}}\normalsize, \large{\frac{5}{{24}}}\normalsize, \large{\frac{6}{{35}}}\normalsize, \large{\frac{7}{{47}}}\normalsize} \right\}\)

Solution.

  1. The set \(R\) descibes \(7\) colors of rainbow. The brown color is superfluous.
  2. The set \(S = \left\{ {2,5,10,13,17,26,37} \right\}\) is described by the expression \({n^2}+1,\) where the number \(n\) varies from \(n = 1\) to \(n = 5.\) The element \(13\) is superfluous.
  3. The set \({P = \left\{ {\text{tetrahedron},\,\text{cube},\,\text{pyramid},}\right.}\) \({\left.{\text{octahedron},\,\text{dodecahedron},}\right.}\) \({\left.{\text{icosahedron}} \right\}}\) lists platonic solids. Pyramid should not be part of this set.
  4. The set \(F = \left\{ {\large{\frac{2}{3}}\normalsize, \large{\frac{3}{8}}\normalsize, \large{\frac{4}{5}}\normalsize, \large{\frac{5}{{24}}}\normalsize, \large{\frac{6}{{35}}}\normalsize, \large{\frac{7}{{47}}}\normalsize} \right\}\) contains fractions of kind \(\large{\frac{n}{{{n^2} – 1}}}\normalsize,\) where \(n\) varies from \(n = 2\) to \(n = 6.\) The fraction \({\large{\frac{7}{{47}}}\normalsize}\) is superfluous.

Example 2.

Write the following sets in roster form:
  1. \(A = \left\{ {n\,|\,n \in \mathbb{Z},\left| n \right| \lt 5} \right\}\)
  2. \(P = \left\{ {p\,|\,p \text{ is prime},\, p \lt 25} \right\}\)
  3. \(B = \left\{ {x \in \mathbb{N}\,|\,{x^2} + 4x – 5 = 0} \right\}\)
  4. \(C = \left\{ {x\,|\,x \in \mathbb{Z},\,{x^2} – 2x – 8 \le 0} \right\}\)
  5. \({D = \bigl\{ {\left( {x,y} \right)\,|\,x \in \mathbb{N}, y \in \mathbb{N},}\bigr.}\) \({\bigl.{{{\left( {x – 1} \right)}^2} + {y^2} \le 1} \bigr\}}\)

Solution.

  1. In roster form, we write down the integer values which satisfy the inequality \(-5 \lt n \lt 5:\)
  2. \[A = \left\{ { – 4, – 3, – 2, – 1,0,1,2,3,4} \right\}.\]
  3. The set \(P = \left\{ {p\,|\,p \text{ is prime},\, p \lt 25} \right\}\) contains prime numbers until \(25:\)
  4. \[P = \left\{ {2,3,5,7,11,13,17,19,23} \right\}.\]
  5. The quadratic equation \({{x^2} + 4x – 5 = 0}\) has the solutions \(x = -5,\) \(x = 1.\) Since the set \(B = \left\{ {x \in \mathbb{N}\,|\,{x^2} + 4x – 5 = 0} \right\}\) contains only natural numbers, it is written in the roster form as
  6. \[B = \{ 1\}.\]
  7. The quadratic inequality \({{x^2} – 2x – 8 \le 0}\) has the solution \(-2 \le x \le 4.\) The set \(C = \left\{ {x\,|\,x \in \mathbb{Z},\,{x^2} – 2x – 8 \le 0} \right\}\) contains integer values in the closed interval \(-2 \le x \le 4,\) so we have
  8. \[C = \left\{ { – 2, – 1,0,1,2,3,4} \right\}.\]
  9. The set \({D = \bigl\{ {\left( {x,y} \right)\,|\,x \in \mathbb{N}, y \in \mathbb{N},}\bigr.}\) \({\bigl.{{{\left( {x – 1} \right)}^2} + {y^2} \le 1} \bigr\}}\) contains points that have integer coordinates and lie inside the circle with radius \(1\) and centered at \(\left({1,0}\right).\)
  10. \[{D = \left\{ {\left( {0,0} \right),\left( {1,0} \right),\left( {2,0} \right),}\right.}\kern0pt{\left.{\left( {1,1} \right),\left( {1, – 1} \right)} \right\}.}\]

Example 3.

How many elements in each of the sets:
  1. \(\varnothing\)
  2. \(\left\{\varnothing\right\}\)
  3. \(\left\{ {\varnothing,\left\{ \varnothing \right\}} \right\}\)
  4. \(\left\{ {\left\{ {\varnothing,\left\{ \varnothing \right\}} \right\}} \right\}\)
  5. \(\left\{ {\varnothing,\left\{ \varnothing \right\},\varnothing} \right\}\)

Solution.

  1. By definition, the empty set \(\varnothing\) contains no elements.
  2. The set \(\left\{\varnothing\right\}\) has \(1\) element – the empty set \(\varnothing.\)
  3. The set \(\left\{ {\varnothing,\left\{ \varnothing \right\}} \right\}\) has \(2\) elements. The first element is the empty set \(\varnothing.\) The second element \(\left\{\varnothing\right\}\) is a set containing, in its turn, the empty set.
  4. The set \(\left\{ {\left\{ {\varnothing,\left\{ \varnothing \right\}} \right\}} \right\}\) contains \(1\) element, which is the set \(\left\{ {\varnothing,\left\{ \varnothing \right\}} \right\}.\)
  5. By definition, a set cannot have duplicate elements. Therefore, the set \(\left\{ {\varnothing,\left\{ \varnothing \right\},\varnothing} \right\}\) has \(2\) elements – \(\varnothing\) and \(\left\{\varnothing\right\}.\)

Example 4.

Determine which of the statements are true and which are not:
  1. \(\varnothing \in \varnothing\)
  2. \(\varnothing \subseteq \varnothing\)
  3. \(\varnothing \in \left\{\varnothing\right\}\)
  4. \(\varnothing \subseteq \left\{\varnothing\right\}\)

Solution.

  1. The statement \(\varnothing \in \varnothing\) is not true as, by definition, the empty set contains no elements.
  2. The statement \(\varnothing \subseteq \varnothing\) is true. The empty set is a subset of every set, including itself.
  3. The statement \(\varnothing \in \left\{\varnothing\right\}\) is true, since the set \(\left\{\varnothing\right\}\) contains one element – \(\varnothing.\)
  4. The statement \(\varnothing \subseteq \left\{\varnothing\right\}\) is true. The empty set is a subset of every set, including the set \(\left\{\varnothing\right\}.\)

Example 5.

Find the power set of the following sets:
  1. \(A = \left\{ {1,\left\{ 1 \right\}} \right\}\)
  2. \(B = \left\{ {1,2,3,3} \right\}\)
  3. \(C = \left\{ {1,\left\{ {2,3} \right\},4} \right\}\)
  4. \(D = \left\{ {x \in \mathbb{Z}\,|\,{x^2} \lt 2} \right\}\)

Solution.

  1. The set \(A = \left\{ {1,\left\{ 1 \right\}} \right\}\) has \(2\) elements. Its power set is given by
  2. \[\mathcal{P}\left( A \right) = \left\{ {\varnothing,\left\{ 1 \right\},\left\{\left\{ 1 \right\}\right\},\left\{ {1,\left\{ 1 \right\}} \right\}} \right\}.\]
  3. The set \(B = \left\{ {1,2,3,3} \right\}\) has \(3\) elements – \(1,\) \(2,\) and \(3.\) The power set of \(B\) includes \(8\) subsets:
  4. \[{\mathcal{P}\left( B \right) = \left\{ {\varnothing,\left\{ 1 \right\},\left\{ 2 \right\},\left\{ 3 \right\},\left\{ {1,2} \right\},\left\{ {2,3} \right\},}\right.}\kern0pt{\left.{\left\{ {1,3} \right\},\left\{ {1,2,3} \right\}} \right\}.}\]
  5. The set \(C = \left\{ {1,\left\{ {2,3} \right\},4} \right\}\) contains elements \(1,\) \(4\) and the set \(\left\{ {2,3} \right\},\) so the power set of \(C\) is written as
  6. \[{\mathcal{P}\left( C \right) = \left\{ {\varnothing,\left\{ 1 \right\},\left\{\left\{ {2,3} \right\}\right\},\left\{ 4 \right\},\left\{ {1,\left\{ {2,3} \right\}} \right\},}\right.}\kern0pt{\left.{\left\{ {1,4} \right\},\left\{ {\left\{ {2,3} \right\},4} \right\},}\right.}\kern0pt{\left.{\left\{ {1,\left\{ {2,3} \right\},4} \right\}} \right\}.}\]
  7. In roster form, the set \(D = \left\{ {x \in \mathbb{Z}\,|\,{x^2} \lt 2} \right\}\) is written as \(D = \left\{ { – 1,0,1} \right\}.\) We see that it has \(3\) elements. Hence,
  8. \[{\mathcal{P}\left( D \right) = \left\{ {\varnothing, \left\{ {-1} \right\},\left\{ 0 \right\},\left\{ 1 \right\},\left\{ { – 1,0} \right\},}\right.}\kern0pt{\left.{\left\{ { – 1,1} \right\},\left\{ {0,1} \right\},}\right.}\kern0pt{\left.{\left\{ { – 1,0,1} \right\}} \right\}.}\]

Example 6.

Let \(A \subseteq B\) and \(a \in A.\) Determine whether these statements are true or false:
  1. \(a \not\in B\)
  2. \(A \in B\)
  3. \(a \subseteq A\)
  4. \(\left\{a\right\} \subseteq A\)
  5. \(\left\{a\right\} \subseteq B\)

Solution.

  1. The statement \(a \not\in B\) is false. Since the set \(A\) is a subset of \(B,\) then each element of \(A\) (including the element \(a\)) belongs to the set \(B,\) that is \(a \in B.\)
  2. The statement \(A \in B\) is false. The \(\in\) symbol defines membership and is related to elements. The set \(A\) is not an element.
  3. Similarly to the previous example, the statement \(a \subseteq A\) is false. The \(\subseteq\) symbol is used for subsets. The element \(a\) is not a subset.
  4. The statement \(\left\{a\right\} \subseteq A\) is true. The set \(\left\{a\right\},\) consisting of one element \(a,\) is a subset of the set \(A.\)
  5. The statement \(\left\{a\right\} \subseteq B\) is true. The set \(\left\{a\right\}\) is a subset of \(A,\) and \(A\) is a subset of \(B.\) The subset relation is transitive. Hence, \(\left\{a\right\}\) is a subset of \(B.\)