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Set operations

Example 1

Let us create sets from the letters of the names of settlements in the United Kingdom. Let the elements of the fundamental set be the letters of the county NORTHUMBERLAND.

Let

the elements of set \latex{ A } be the letters of BARNET;
the elements of set \latex{ B } be the letters of LUTON;
the elements of set \latex{ C } be the letters of LAMBETH.

Let us create a set diagram and write the letters into the corresponding parts of the sets.

List the letters which are

not in set \latex{ C };
in both set \latex{ A } and set \latex{ B };
in all three sets;
in set \latex{ C }, but not in set \latex{ A };
not in any of sets \latex{ A }, \latex{ B } and \latex{ C }.

What can we say about the letters of the town BOURNEMOUTH?

Solution (a)

Every element can be in a set only once, thus we cannot write the same letter twice. (Figure 14)

Solution (b)

The letters of the fundamental set which are not in set C: R; N; U; O; D.
The letters which are in both set A and set B: N and T.
Only the letter T is in all three sets.
The letters of set C which are not in set A: L; M; H.
The letters of the fundamental set which is not in any of sets A, B and C: D.

Solution (c)

The letters of the town BOURNEMOUTH are in at least one of sets A, B and C.

⯁ ⯁ ⯁

In Example 1 we created new sets out of the elements of other sets, i.e. we did set operations.

Henceforth let us determine these operations precisely and let us find a few more examples.

DEFINITION: When examining sets a set can be given all the examined sets that are subsets of it; it is called fundamental set or universe. Notation: U

DEFINITION: The set of those elements of the fundamental set, which are not elements of set A are called the complement set or complement of set A. Notation: \latex{\overline{A}}

\latex{\overline{A}=\left\{x\mid x\notin A\right\} ;\;\overline{\overline{A}}=A;\;\overline{U}=\varnothing \;}and\latex{\;\overline{\varnothing}=U}

For example let the fundamental set be U = {\latex{10} - \latex{30}, integers}, and the examined sets be:

R = {10–30, even numbers};
S = {10–30, numbers divisible by 3};
T = {10–30, numbers divisible by 5}.

Then the complements of sets R, S, T are the following:

\latex{\overline{R}} = {10–30, odd numbers};
\latex{\overline{S}} = {10–30, integers not divisible by 3};
\latex{\overline{T}} = {10–30, integers not divisible by 5}. (Figure 23)

Henceforth the examples refer to these sets U, R, S, T.

⯁ ⯁ ⯁

DEFINITION: The intersection of two sets is the set of all elements which are elements of both sets. Notation: \latex{\cap} (Figure 16)

\latex{A \cap B=\lbrace x\mid x\in A} and \latex{x\in B\rbrace}

For example: S \latex{\cap} T = {15; 30} = {10–30, numbers divisible by 3 and 5} (Figure 23)

The properties of the intersection operation for arbitrary sets A, B, C:

commutative (permutable): \latex{A\cap B} = \latex{B\cap A};
associative (groupable): \latex{A\cap \left(B\cap C\right) =\left(A\cap B\right) \cap C}; (Figure 17)
\latex{A\cap A=A};\latex{\;A\cap \varnothing =\varnothing}; \latex{\;A\cap U=A}.

DEFINITION: Two sets are disjoint, if they do not have any elements in common, i.e. their intersection is the empty set: \latex{A\cap B =\varnothing}. (Figure 18)

For example: sets K = {12; 15} and L = {23; 28} are disjoint.

DEFINITION: The union of two sets is the set of all elements which are elements of at least one of the sets. Notation: \latex{\cup } (Figure 19)

\latex{A \cup B=\lbrace x\mid x\in A} or \latex{x\in B\rbrace}

For example: \latex{S\cup T} = {10–30, numbers divisible by 3 or 5} = {10; 12; 15; 18; 20; 21; 24; 25; 27; 30}. (Figure 23)

The properties of the union operation for arbitrary sets A, B, C:

commutative: \latex{A\cup B=B\cup A};
associative: \latex{A\cup \left(B\cup C\right) =\left(A\cup B\right) \cup C}; (Figure 20)
\latex{A\cup A=A}; \latex{A\cup\varnothing =A}; \latex{A\cup U =U}.

⯁ ⯁ ⯁

The union is distributive over the intersection:

\latex{A\cup \left(B\cap C\right) =\left(A\cup B\right) \cap \left(A\cup C\right) }. (Figure 21)

For example: R \latex{\cup} (S \latex{\cap} T) = {10–30, even numbers} \latex{\cup} {15; 30} = {10; 12; 14; 15; 16; 18; 20; 22; 24; 26; 28; 30}. (Figure 23)

(R \latex{\cup} S) \latex{\cap} (R \latex{\cup} T) = {10–30, even numbers or numbers divisible by 3} \latex{\cap} {10–30, even numbers or numbers divisible by 5} = {10; 12; 14; 15; 16; 18; 20; 22; 24; 26; 28; 30}. (Figure 23)

Figure 21

\latex{ U }

\latex{ A }

\latex{C }

\latex{ B }

Figure 22

\latex{ U }

\latex{ A }

\latex{C }

\latex{ B }

\latex{A\cup \left(B\cap C\right) =\left(A\cup B\right) \cap \left(A\cup C\right) }

\latex{A\cap \left(B\cup C\right) =\left(A\cap B\right) \cup \left(A\cap C\right) }

The intersection is distributive over the union:

\latex{A\cap \left(B\cup C\right) =\left(A\cap B\right) \cup \left(A\cap C\right) }. (Figure 22)

For example:
R \latex{\cap} (S \latex{\cup} T) = {10–30, even numbers} \latex{\cap} {10–30, numbers divisible by 3 or 5} = {10; 12; 14; 16; 18; 20; 22; 24; 26; 28; 30} \latex{\cap} {10; 12; 15; 18; 20; 21; 24; 25; 27; 30} = {10–30, numbers divisible by 6 or 10} = {10; 12; 18; 20; 24; 30}. (Figure 23)

(R \latex{\cap} S) \latex{\cup} (R \latex{\cap} T)={10–30, numbers divisible by 6} \latex{\cup} {10–30, numbers divisible by 10}={10; 12; 18; 20; 24; 30}. (Figure 23)

⯁ ⯁ ⯁

DEFINITION: The difference of sets A and B is the set of those elements of set A which are not elements of set B. Notation: A\B (Figure 24)

\latex{A\setminus B=\lbrace x\mid x\in A\;}and\latex{\;x\notin B\rbrace}; \latex{A\setminus B=A\cap \overline{B}}

For example: \latex{R\setminus S}={\latex{10} – \latex{30}, numbers divisible by \latex{2} but not divisible by \latex{3}}=
={\latex{10; 14; 16; 20; 22; 26; 28}}. (Figure 23)

The properties of the difference operation for arbitrary set A:

\latex{A\setminus A=\varnothing ;\;\;A\setminus \varnothing =A;\;\;\varnothing \setminus A=\varnothing ;\;\;U\setminus A=\overline{A}}.

Example 2

The Venn diagram which contains one set divides the fundamental set into two parts. Two intersecting sets divide the fundamental set into four parts. Into how many parts do three sets divide the fundamental set if we represent them on a Venn diagram? Let us draw and characterise each part with the help of set operations.

Solution

Each new circle divides the already existing parts into two, thus it doubles the number of parts; i.e. \latex{3} circles divide the fundamental set into a maximum of 2 × 2 × 2 = 8 parts. In Figure 25 we divided the fundamental set into exactly \latex{8} parts.

1. \latex{\overline{A}\cap \overline{B}\cap \overline{C}}

2. \latex{A\cap \overline{B}\cap \overline{C}}

3. \latex{\overline{A}\cap B\cap \overline{C}}

4. \latex{\overline{A}\cap \overline{B}\cap C}

5. \latex{A\cap B\cap \overline{C}}

6. \latex{\overline{A}\cap B\cap C}

7. \latex{A\cap \overline{B}\cap C}

8. \latex{A\cap B\cap C}

Figure 25

Example 3

Let us define the sets \latex{\overline{S}\cup \overline{T}; \overline{S\cap T};\overline{S}\cap \overline{T};\overline{S\cup T}}, if \latex{U} = {10–30, integers}; \latex{S} = {10–30, numbers divisible by 3}; \latex{T} = {10–30, numbers divisible by 5}.

Solution

\latex{\overline{S}\cup \overline{T}} = {10–30, integers not divisible by 3} \latex{\cup} {10–30, integers not divisible by 5} = {10–30, integers which are not divisible by either 3 or 5, i.e. by 15} = {10; 11; 12; 13; 14; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29}. (Figure 27)

\latex{\overline{S\cap T}} = {10–30, integers, except for numbers divisible by 3 and 5} = {10; 11; 12; 13; 14; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29}. (Figure 27)

\latex{\overline{S}\cap \overline{T}} = {10–30, numbers not divisible by 3} \latex{\cap} {10–30, numbers not divisible by 5} = {10–30, integers not divisible by either 3 or 5} = {10; 11; 13; 14; 16; 17; 19; 22; 23; 26; 28; 29}. (Figure 29)

\latex{\overline{S\cup T}} = {10–30, integers which are not divisible by 3 or 5} = {10–30, integers which are not divisible by either 3 or 5} = {10; 11; 13; 14; 16; 17; 19; 22; 23; 26; 28; 29}. (Figure 29)

Based on the above example we can assume that the two equalities – also known as De Morgan's laws – are true for any two sets A, B (Figure 28):

\latex{\overline{A}\cup \overline{B}=\overline{A\cap B}\;}and \latex{\;\overline{A}\cap \overline{B}=\overline{A\cup B}}

\latex{\overline{A}\cup \overline{B}=\overline{A\cap B}}

\latex{ U }

\latex{ A }

\latex{ B }

\latex{ A }

\latex{\overline{A}\cap \overline{B}=\overline{A\cup B}}

\latex{ U }

\latex{ A }

\latex{ B }

\latex{ A }

Figure 28

Figure 29

\latex{\overline{S}\cap \overline{T}=\overline{S\cup T}}

\latex{ U }

\latex{T}

\latex{S}

\latex{29}

\latex{11}

\latex{13}

\latex{14}

\latex{16}

\latex{17}

\latex{19}

\latex{12}

\latex{22}

\latex{23}

\latex{26}

\latex{28}

\latex{10}

\latex{20}

\latex{25}

\latex{15}

\latex{30}

\latex{27}

\latex{24}

\latex{21}

\latex{18}

Exercises

{{exercise_number}}. Give \latex{\overline{A} , \overline{B}}, \latex{A\cap B} and \latex{A\cup B} for the following sets.

\latex{A = \left\{2; 4; 6; 8\right\}} and \latex{B = \left\{1; 3; 4; 6; 7\right\}}, if \latex{U = \left\{1; 2; 3; 4; 5; 6; 7; 8; 9; 10\right\}}
\latex{A = \left\{a;b;c\right\}} and \latex{B = \left\{d;e;f\right\}}, if \latex{U = \left\{a;b;c;d;e;f\right\}}
\latex{A = \left\{a;e;i;o;u\right\}} and \latex{B = \left\{e;o;u\right\}}, if \latex{U = \left\{a;d;e;i;o;u\right\}}
\latex{A = \left\{o;b;t;a;i;n\right\}} and \latex{B = \left\{r;a;i;n\right\}}, if \latex{U = \left\{o;b;t;a;i;n;r;e;z\right\}}
\latex{A = \left\{5;10;15;...\right\}} and \latex{B = \left\{10;20;30;...\right\}}, if \latex{U = \N^+}
\latex{A = \left\{1;3;5;7;...\right\}} and \latex{B = \left\{2;4;6;8;...\right\}}, if \latex{U = \N^+}
A = {numbers divisible by \latex{6}} and B = {numbers divisible by \latex{4}}, if \latex{U = \N}
A = {numbers divisible by \latex{15}} and B = {numbers divisible by \latex{6}}, if \latex{U = \N}
A = {rectangles} and B = {rhombi}, if U = {parallelograms}
A = {kites} and B = {parallelograms}, if U = {quadrilaterals}
A = {even numbers} and B = {odd numbers}, if \latex{U = \Z}
A = {rational numbers} and B = {irrational numbers}, if \latex{U = \R}
A = {positive numbers} and B = {negative numbers}, if \latex{U = \R}
A = {non-negative numbers} and B = {non-positive numbers}, if \latex{U = \R}

{{exercise_number}}. Define set \latex{ A } if \latex{U=\N} and

a) \latex{\overline{A}=\lbrace n\mid n\in \N} and n is at most \latex{ 15\rbrace };
b) \latex{\overline{A}=\lbrace n\mid n\in \N} and n is at least \latex{ 30\rbrace }.

{{exercise_number}}. Let \latex{U=\left\{1;2;3;4;5;6;7\right\}}, \latex{A=\left\{2;4;6;7\right\}} and \latex{B=\left\{1;3;5;6;7\right\}}. Give the following sets and represent them on a Venn diagram.

a) \latex{\overline{A}}

b) \latex{\overline{B}}

c) \latex{\overline{A}\cup B}

d) \latex{\overline{A}\cap B}

e) \latex{\overline{A}\cup \overline{B}}

f) \latex{A\cup \overline{B}}

g) \latex{A\cap \overline{B}}

h) \latex{\overline{A}\cap \overline{B}}

i) \latex{\overline{A\cap B}}

j) \latex{\overline{A\cup B}}

k) \latex{A\setminus B}

l) \latex{\overline{A}\setminus B}

m) \latex{A\setminus \overline{B}}

n) \latex{\overline{A}\setminus \overline{B}}

o) \latex{\overline{A\setminus B}}

p) \latex{A\cup U}

q) \latex{B\cap U}

r) \latex{A\setminus U}

{{exercise_number}}. Let \latex{U=\left\{0; 1; 2; 3; 4; 5; 6; 7; 8; 9\right\} }, \latex{A=\left\{1; 3; 4; 5; 7\right\} }, \latex{B=\left\{2; 3; 4; 5; 6\right\} } and \latex{C=\left\{0; 2; 4; 6; 8; 9\right\} }.

Give the following sets and represent them on a Venn diagram.

a) \latex{\overline{A}\cap \overline{B}}

b) \latex{\overline{A}\cap B}

c) \latex{\overline{A\cap B}}

d) \latex{\overline{A\cup B}}

e) \latex{\overline{A}\cup \overline{B}}

f) \latex{A\cup \overline{B}}

g) \latex{A\cap \overline{B}}

h) \latex{A\cup C}

i) \latex{B\cap C}

j) \latex{\overline{B}\cup C}

k) \latex{\left(A\cap B\right) \cup C}

l) \latex{\left(A\cup B\right) \cap C}

m) \latex{\left(\overline{A}\cap B\right) \cup C}

n) \latex{\left(\overline{A}\cup B\right) \cap \overline{C}}

o) \latex{\overline{A\cup B} \cap \overline{C}}

p) \latex{\overline{A}\cap\overline{B}\cap\overline{C}}

{{exercise_number}}. Let \latex{U=\left\{a; e; i; o; u\right\}} , \latex{A=\left\{i; o; u\right\}} and \latex{B=\left\{e;i;o\right\}} and \latex{C=\left\{a;i;o\right\}}. Which of thefollowing statements are true and which are false?

a) \latex{A\cap B=C}

b) \latex{B\cup C=U}

c) \latex{\left(A\cap B\right) \subset C}

d) \latex{A\cap B\cap C=\varnothing }

e) \latex{\left(B\cap C\right) \subseteq A}

f) \latex{\overline{A}\cap \overline{B}=\varnothing }

g) \latex{\overline{A}\cap \overline{B}= \overline{A\cup B}}

h) \latex{\overline{A}\cup \overline{B}= \overline{A\cap B}}

{{exercise_number}}. Define sets \latex{ A } and \latex{ B } if we know that:

a) \latex{A\cup B=\left\{5;6;7,8;9;10\right\} },

\latex{A\setminus B=\left\{8;9;10\right\}} and

\latex{A\cap B=\left\{5\right\}}.

b) \latex{A\cup B=\left\{5;6;7;8;9;10\right\} },
\latex{B\setminus A=\left\{5;6;8;9;10\right\}} and
\latex{A\cap B=\varnothing}.

{{exercise_number}}. Define sets \latex{ A, B, C }, if we know that:

a) \latex{A\cup B\cup C=\left\{1;2;3;4;5;6;7;8;9;10\right\}}, \latex{A\cup B=\left\{1;2;3;6;7;8;9;10\right\}}, \latex{A\setminus B=\left\{8;9;10\right\}}, \latex{A\cap B=\left\{1;2;3\right\}}, \latex{A\cap B\cap C=\left\{1;3\right\}}, \latex{A\cap C=\left\{1;3;8\right\}} and \latex{B\setminus C=\left\{2;7\right\}};

b) \latex{A\cup B\cup C=\left\{1;2;3;4;5;6;7;8;9;10\right\}}, \latex{A\cap B=\left\{1;2;5;6;7\right\}}, \latex{A\cap C=\left\{5;6;7;8\right\}}, \latex{B\cap C=\left\{3;5;6;7;\right\}}, \latex{A\cap B\cap C=\left\{5;6;7\right\}}, \latex{A\cup B={\left\{1;2;3;4;5;6,7;8;9\right\}}} and \latex{B\setminus A=\left\{3;9\right\}}.

{{exercise_number}}. Let set \latex{ A } be the set of girls attending the class, set \latex{ B } be the set of left-handed pupils attending the class. Create a set diagram and place the pupils of the class.

{{exercise_number}}. Let set \latex{ A } be the set of pupils of the class younger than \latex{15}, set \latex{ B } be the set of pupils wearing glasses, and set \latex{ C } be the set of pupils taller than \latex{160} \latex{cm}. Create a set diagram and place the pupils of the class.

{{exercise_number}}. Every piano player also attends the school choir, and every school choir member has long hair. There is a school choir member who got an \latex{ A } in Mathematics. Which of the listed statements are true for sure?

Every piano player attends the school choir and has long hair.
Not every piano player got an \latex{ A } in Mathematics.
Every piano player has long hair and got an \latex{ A } in Mathematics.
There is a pupil who got an \latex{ A } in Mathematics and has long hair.

{{exercise_number}}. Every friend of the red owls wears shorts. There is a green elephant who is a friend of the red owls. Which of the following statements are true and which are false?

There is no green elephant who wears shorts.
If someone wears shorts then he/she is a friend of the red owls.
If someone does not wear shorts then he/she is a green elephant.
If someone does not wear shorts then he/she is not a friend of the red owls.

{{exercise_number}}. We know that every \latex{ X } is a \latex{ Y } too, but only some of the \latex{ Y }s will be \latex{ Z }. Which of the following statements are true for sure?

There is no \latex{ X } which is a \latex{ Z }.
If something is not a \latex{ Y }, then it is not an \latex{ X }.
If something is not a \latex{ Z }, then it is not an \latex{ X }.

{{exercise_number}}.

a) A square is given. In its centre we affix one vertex of a congruent square. What is the area of the intersection of the two squares if the area of the original square is \latex{8\, cm^2?} What is the difference of the areas coloured in yellow and blue?

b) A regular hexagon is given. In its centre we affix one vertex of a congruent hexagon. What is the area of the intersection of the two hexagons if the area of the original hexagon is \latex{12\,cm^2?} What is the difference of the areas coloured in yellow and blue?

Puzzle

\latex{30} pupils attend our class, \latex{18} of them learn German and \latex{17} of them learn French. But \latex{18} \latex{ + } \latex{17} \latex{ \gt } \latex{30}.
Where is the error?

Mathematics 9.

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