Calculus

Set Theory

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Introduction to Sets

Solved Problems

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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. \({R = \left\{ {\text{red},\,\text{orange},\,\text{yellow},}\right.}\) \({\left.{\text{brown},\text{green},\text{blue},}\right.}\) \({\left.{\text{indigo},\text{violet}} \right\}}\)
  2. \(S = \left\{ {2,5,10,13,17,26,37} \right\}\)
  3. \({P = \left\{ {\text{tetrahedron},\,\text{cube},}\right.}\) \({\left.{\text{pyramid},\text{octahedron},}\right.}\) \({\left.{\text{dodecahedron},}\right.}\) \({\left.{\text{icosahedron}} \right\}}\)
  4. \(F = \left\{ {\frac{2}{3}, \frac{3}{8}, \frac{4}{5}, \frac{5}{{24}}, \frac{6}{{35}}, \frac{7}{{47}}} \right\}\)

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. \(C = \left\{ {x\,|\,x \in \mathbb{Z},}\right.\) \(\left.{{x^2} - 2x - 8 \le 0} \right\}\)
  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. \({R = \left\{ {\text{red},\,\text{orange},\,\text{yellow},}\right.}\) \({\left.{\text{brown},\text{green},\text{blue},}\right.}\) \({\left.{\text{indigo},\text{violet}} \right\}}\)
  2. \(S = \left\{ {2,5,10,13,17,26,37} \right\}\)
  3. \({P = \left\{ {\text{tetrahedron},\,\text{cube},}\right.}\) \({\left.{\text{pyramid},\text{octahedron},}\right.}\) \({\left.{\text{dodecahedron},}\right.}\) \({\left.{\text{icosahedron}} \right\}}\)
  4. \(F = \left\{ {\frac{2}{3}, \frac{3}{8}, \frac{4}{5}, \frac{5}{{24}}, \frac{6}{{35}}, \frac{7}{{47}}} \right\}\)

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 \(F = \left\{ {\frac{2}{3}, \frac{3}{8}, \frac{4}{5}, \frac{5}{{24}}, \frac{6}{{35}}, \frac{7}{{47}} } \right\}\) contains fractions of kind \(\frac{n}{{{n^2} - 1}},\) where \(n\) varies from \(n = 2\) to \(n = 6.\) The fraction \({\frac{7}{{47}} }\) 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. \(C = \left\{ {x\,|\,x \in \mathbb{Z},}\right.\) \(\left.{{x^2} - 2x - 8 \le 0} \right\}\)
  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 \(-5 \lt n \lt 5:\)
    \[A = \left\{ { - 4, - 3, - 2, - 1,0,1,2,3,4} \right\}.\]
  2. The set \(P = \left\{ {p\,|\,p \text{ is prime},\, p \lt 25} \right\}\) contains prime numbers until \(25:\)
    \[P = \left\{ {2,3,5,7,11,13,17,19,23} \right\}.\]
  3. 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
    \[B = \{ 1\}.\]
  4. 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
    \[C = \left\{ { - 2, - 1,0,1,2,3,4} \right\}.\]
  5. 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 \(\left({1,0}\right).\)
    \[D = \left\{ {\left( {0,0} \right),\left( {1,0} \right),\left( {2,0} \right), \left( {1,1} \right),\left( {1, - 1} \right)} \right\}.\]

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
    \[\mathcal{P}\left( A \right) = \left\{ {\varnothing,\left\{ 1 \right\},\left\{\left\{ 1 \right\}\right\},\left\{ {1,\left\{ 1 \right\}} \right\}} \right\}.\]
  2. The set \(B = \left\{ {1,2,3,3} \right\}\) has \(3\) elements - \(1,\) \(2,\) and \(3.\) The power set of \(B\) includes \(8\) subsets:
    \[\mathcal{P}\left( B \right) = \left\{ {\varnothing,\left\{ 1 \right\},\left\{ 2 \right\},\left\{ 3 \right\},\left\{ {1,2} \right\},\left\{ {2,3} \right\},\left\{ {1,3} \right\},\left\{ {1,2,3} \right\}} \right\}.\]
  3. 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
    \[\mathcal{P}\left( C \right) = \left\{ {\varnothing,\left\{ 1 \right\},\left\{\left\{ {2,3} \right\}\right\},\left\{ 4 \right\},\left\{ {1,\left\{ {2,3} \right\}} \right\},\left\{ {1,4} \right\},\left\{ {\left\{ {2,3} \right\},4} \right\},\left\{ {1,\left\{ {2,3} \right\},4} \right\}} \right\}.\]
  4. 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,
    \[\mathcal{P}\left( D \right) = \left\{ {\varnothing, \left\{ {-1} \right\},\left\{ 0 \right\},\left\{ 1 \right\},\left\{ { - 1,0} \right\},\left\{ { - 1,1} \right\},\left\{ {0,1} \right\},\left\{ { - 1,0,1} \right\}} \right\}.\]

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 \(B.\)
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