# Introduction to Sets

## 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.

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

### Example 2

Write the following sets in roster form:

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

### Example 3

How many elements in each of the sets:

- \(\varnothing\)
- \(\left\{\varnothing\right\}\)
- \(\left\{ {\varnothing,\left\{ \varnothing \right\}} \right\}\)
- \(\left\{ {\left\{ {\varnothing,\left\{ \varnothing \right\}} \right\}} \right\}\)
- \(\left\{ {\varnothing,\left\{ \varnothing \right\},\varnothing} \right\}\)

### Example 4

Determine which of the statements are true and which are not:

- \(\varnothing \in \varnothing\)
- \(\varnothing \subseteq \varnothing\)
- \(\varnothing \in \left\{\varnothing\right\}\)
- \(\varnothing \subseteq \left\{\varnothing\right\}\)

### Example 5

Find the power set of the following sets:

- \(A = \left\{ {1,\left\{ 1 \right\}} \right\}\)
- \(B = \left\{ {1,2,3,3} \right\}\)
- \(C = \left\{ {1,\left\{ {2,3} \right\},4} \right\}\)
- \(D = \left\{ {x \in \mathbb{Z}\,|\,{x^2} \lt 2} \right\}\)

### Example 6

Let \(A \subseteq B\) and \(a \in A.\) Determine whether these statements are true or false:

- \(a \not\in B\)
- \(A \in B\)
- \(a \subseteq A\)
- \(\left\{a\right\} \subseteq A\)
- \(\left\{a\right\} \subseteq B\)

### Example 1.

In every set, all elements, except one, have some property. Describe this property and find the element that does not have it.

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

Solution.

- The set \(R\) descibes \(7\) colors of rainbow. The brown color is superfluous.
- 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.
- 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.
- 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:

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

Solution.

- In roster form, we write down the integer values which satisfy the inequality \(-5 \lt n \lt 5:\)
\[A = \left\{ { - 4, - 3, - 2, - 1,0,1,2,3,4} \right\}.\]
- The set \(P = \left\{ {p\,|\,p \text{ is prime},\, p \lt 25} \right\}\) contains prime numbers until \(25:\)
\[P = \left\{ {2,3,5,7,11,13,17,19,23} \right\}.\]
- 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
\[B = \{ 1\}.\]
- 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
\[C = \left\{ { - 2, - 1,0,1,2,3,4} \right\}.\]
- 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:

- \(\varnothing\)
- \(\left\{\varnothing\right\}\)
- \(\left\{ {\varnothing,\left\{ \varnothing \right\}} \right\}\)
- \(\left\{ {\left\{ {\varnothing,\left\{ \varnothing \right\}} \right\}} \right\}\)
- \(\left\{ {\varnothing,\left\{ \varnothing \right\},\varnothing} \right\}\)

Solution.

- By definition, the empty set \(\varnothing\) contains no elements.
- The set \(\left\{\varnothing\right\}\) has \(1\) element - the empty set \(\varnothing.\)
- 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.
- The set \(\left\{ {\left\{ {\varnothing,\left\{ \varnothing \right\}} \right\}} \right\}\) contains \(1\) element, which is the set \(\left\{ {\varnothing,\left\{ \varnothing \right\}} \right\}.\)
- 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:

- \(\varnothing \in \varnothing\)
- \(\varnothing \subseteq \varnothing\)
- \(\varnothing \in \left\{\varnothing\right\}\)
- \(\varnothing \subseteq \left\{\varnothing\right\}\)

Solution.

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

- \(A = \left\{ {1,\left\{ 1 \right\}} \right\}\)
- \(B = \left\{ {1,2,3,3} \right\}\)
- \(C = \left\{ {1,\left\{ {2,3} \right\},4} \right\}\)
- \(D = \left\{ {x \in \mathbb{Z}\,|\,{x^2} \lt 2} \right\}\)

Solution.

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

- \(a \not\in B\)
- \(A \in B\)
- \(a \subseteq A\)
- \(\left\{a\right\} \subseteq A\)
- \(\left\{a\right\} \subseteq B\)

Solution.

- 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 \(a \in B.\)
- The statement \(A \in B\) is false. The \(\in\) symbol defines membership and is related to elements. The set \(A\) is not an element.
- 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.
- 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 \(A.\)
- 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 \(B.\)