# 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.

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 $$B.$$