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Sets
Example 1
The head of the intelligence agency received reports about the observed people in the form of a set diagram. Based on this what could he figure out about \latex{ Dr. \,X, } \latex{ Dr.\,Y }and \latex{ Dr.\,Z }?
women\latex{ Dr.\,Z }\latex{ Dr.\,Y }\latex{ Dr.\,X }wear glassesspiestall
Solution
\latex{ Dr. \,X }: Spy, who is not tall, is a man not wearing glasses.
\latex{ Dr. \,Y }: Spy, who wears glasses, is not tall and is a woman.
\latex{ Dr.\, Z }: Spy, who is tall, is a man not wearing glasses.
\latex{ Dr. \,Y }: Spy, who wears glasses, is not tall and is a woman.
\latex{ Dr.\, Z }: Spy, who is tall, is a man not wearing glasses.
⯁ ⯁ ⯁
The expressions “set” and “element of a set” are basic concepts, i.e. they are not defined only paraphrased, demonstrated with the help of examples.
We use the word set for a collection of objects, and the objects in the collection are the elements of the set. We can decide whether an arbitrary object belongs to a set or not.
Notations: Generally sets are denoted by capital letters: \latex{ A }, \latex{ B }, \latex{ C }.
We use the word set for a collection of objects, and the objects in the collection are the elements of the set. We can decide whether an arbitrary object belongs to a set or not.
Notations: Generally sets are denoted by capital letters: \latex{ A }, \latex{ B }, \latex{ C }.
\latex{x} is an element of set \latex{ A }: \latex{x} \latex{\in} \latex{ A }.
\latex{x} is not an element of set \latex{ A }: \latex{x} \latex{\notin}\latex{ A }.
Set \latex{ A } consists of elements \latex{a}; \latex{b}; \latex{c} : \latex{ A }={\latex{a; b; c}}.
\latex{x} is not an element of set \latex{ A }: \latex{x} \latex{\notin}\latex{ A }.
Set \latex{ A } consists of elements \latex{a}; \latex{b}; \latex{c} : \latex{ A }={\latex{a; b; c}}.
For example the set of the vowels of the English alphabet:
E = {a; e; i; o; u}.
We list every element of the set only once and the order of the elements
does not matter, i.e. {a; e; i; o; u} = {a; u; i; e; o}.
E = {a; e; i; o; u}.
We list every element of the set only once and the order of the elements
does not matter, i.e. {a; e; i; o; u} = {a; u; i; e; o}.
Example 2
\latex{ }Let us consider the below sets \latex{ A }, \latex{ B }, \latex{ C } and\latex{ D }.
– Let the elements of set \latex{ A } be the two-digit numbers divisible by \latex{9}.
– Let the elements of set \latex{ B } be the two-digit even numbers.
– Let the elements of set \latex{ C } be those two-digit numbers which only consist of even digits.
– Let the elements of set \latex{ B } be the two-digit even numbers.
– Let the elements of set \latex{ C } be those two-digit numbers which only consist of even digits.
– Let the elements of set \latex{ D } be those two-digit numbers where the sum of the
digits is \latex{9}.
digits is \latex{9}.
- Give the number of the elements of sets \latex{ A }, \latex{ B }, \latex{ C } and \latex{ D }.
- How many two-digit numbers are there which are elements of both set \latex{ A }and set \latex{ C }?
Solution (a)
\latex{ A } = {\latex{18; 27; 36; 45; 54; 63; 72; 81; 90; 99}}, so the order of set \latex{ A } is \latex{10}.
As there are \latex{90} two-digit numbers and every second one is even,
the number of two-digit even numbers is \latex{45}; i.e. the order of set \latex{ B } is \latex{45}.
The elements of set \latex{ C } are those even numbers, where there are even digits also on the tens' place. \latex{4} different digits can (\latex{0} cannot) appear in the tens' place and there are \latex{5} even numbers for every ten. So there are \latex{20} two-digit numbers for which both digits are even.
The elements of set \latex{ D } are divisible by \latex{9}. The sole two-digit number divisible by \latex{9} where the sum of the digits is not \latex{9} is \latex{99}. So there are \latex{9} two-digit numbers for which the sum of the digits is \latex{9}.
Solution (b)
For those two-digit numbers which are elements of both set \latex{ A } and set \latex{ C } it should be fulfilled at the same time that they are divisible by \latex{9} and both of their digits are even. From the list of the elements of set \latex{ A } we can see that there is no such number. Therefore there are \latex{0} such elements which are elements of both set \latex{ A } and set \latex{ C }.
DEFINITION: The set with \latex{0} elements is called the empty set, notation: \latex{\varnothing} or { }.
Note: D = {\latex{0}} is a set with one element and this element is number \latex{0}.
E = {\latex{\varnothing}} is a set with one element and this element is the empty set.
The methods of describing sets:
(1) With a rule or property that uniquely defines the elements of the set.
For example: \latex{C=\lbrace}numbers not greater than \latex{4}, divisible by \latex{2 \rbrace = \lbrace x \space \vert \space x \leq 4} and \latex{x} is divisible by \latex{2 \rbrace}, i.e. set \latex{C} has those \latex{x} numbers as elements which are not greater than \latex{4} and are divisible by \latex{2}.
(1) With a rule or property that uniquely defines the elements of the set.
For example: \latex{C=\lbrace}numbers not greater than \latex{4}, divisible by \latex{2 \rbrace = \lbrace x \space \vert \space x \leq 4} and \latex{x} is divisible by \latex{2 \rbrace}, i.e. set \latex{C} has those \latex{x} numbers as elements which are not greater than \latex{4} and are divisible by \latex{2}.
(2) By listing the elements of the set.
For example: \latex{ G } = {\latex{2; 7; 8}}. It can happen that too many elements should be listed. In cases like this the obvious regularity is denoted by three dots, a so called “ellipsis”, e.g. set
\latex{ H } = {\latex{5; 10; 15; … ; 90; 95; 100}} is the set of positive integers not greater than \latex{100} and divisible by \latex{5}, set \latex{\N} = {\latex{0; 1; 2; 3; …}} is the set of natural numbers.
\latex{ H } = {\latex{5; 10; 15; … ; 90; 95; 100}} is the set of positive integers not greater than \latex{100} and divisible by \latex{5}, set \latex{\N} = {\latex{0; 1; 2; 3; …}} is the set of natural numbers.
DEFINITION: We say that two sets are equal if the two sets have the same elements. i.e. \latex{ A } = \latex{ B } , if for all \latex{x\in A} it is also true that \latex{x\in B}, and the other way round, for all \latex{y \in B} it is also true that \latex{y\in A}.
We can represent the sets with the help of Venn diagrams.
For example the following can be represented as in Figure 8:
– the set of positive integers: \latex{\N^+};
– the set of natural numbers: \latex{\N};
– the set of integers: \latex{\Z};
– the set of rational numbers: \latex{\Q};
– the set of irrational numbers: \latex{\char} or \latex{\Q^*};
– the set of real numbers: \latex{\R}.
– the set of positive integers: \latex{\N^+};
– the set of natural numbers: \latex{\N};
– the set of integers: \latex{\Z};
– the set of rational numbers: \latex{\Q};
– the set of irrational numbers: \latex{\char} or \latex{\Q^*};
– the set of real numbers: \latex{\R}.
The figures corresponding to the sets are all labelled and there are also some examples for the elements of each set.
Figure 8\latex{I}\latex{\Q}\latex{\Z}\latex{\N}\latex{\N^+}\latex{\R}\latex{ \sqrt{2} }\latex{ -0.61 }\latex{ -3}\latex{ -1}\latex{ 0}\latex{ -826}\latex{ 1}\latex{ 947}\latex{ 0.23}\latex{ 1/3}\latex{ π}DEFINITION: We say that set \latex{ A } is a subset of set B, if all the elements of set \latex{ A } are also elements of set B. Notation: \latex{A \subseteq B} (Figure 9)
DEFINITION: Set A is a real subset of set B, if set A is a subset of set B and there is an element of set B which is not an element of set A. Notation: \latex{A \subset B}
Figure 9\latex{ U }\latex{ A }\latex{ B }For example: {even numbers} \latex{\subset } {integers}.
Every set is a subset of itself: \latex{A \subseteq A }.
The empty set is a subset of every set: \latex{\varnothing \subseteq A}.
The empty set is a subset of every set: \latex{\varnothing \subseteq A}.
There can be found such set A and set B for which none of the following is true:
A \latex{\subseteq } B, A = B, B \latex{\subseteq } A. For example for the two sets represented by the Venn diagram in Figure 10 none of the relations are true. Similarly good examples are set A = {1; 2; 3; 4; 5; 6} and set B = {3; 4; 5; 6; 7; 8}.
subset
real subset
real subset
Figure 10\latex{ U }\latex{ B }\latex{ A }Example 3
List all the subsets of set \latex{ {a; b; c}. }
Solution
\latex{ \left\{a;b\right\}};\latex{ \left\{a;c\right\}};\latex{ \left\{b;c\right\} }0 elements1 elements2 elements3 elements\latex{ \left\{a;b;c\right\} }\latex{ \left\{a\right\}};\latex{ \left\{b\right\}};\latex{ \left\{c\right\} }Set {a; b; c} with \latex{3} elements has a total of \latex{8} subsets. From the list of the subsets it can be seen that every set with \latex{3} elements has \latex{2^3} subsets. The empty set has one subset: itself.
LexiRexiFoxiMaxiExample 4
For which points of the plane is it true that they are the following distance away from a given point A:
- \latex{2} \latex{cm};
- at most \latex{2} \latex{cm};
- at least \latex{1} \latex{cm} but at most \latex{2} \latex{cm}?
Solution
The solutions can be seen in Figure 11 in blue, red and green colour.
\latex{ A }a)b)c)\latex{ A }\latex{ A }Figure 11
DEFINITION: Two sets are equivalent if a one-to-one correspondence (bijection) can be established between them.
For example set A = {1; 2; 3} is equivalent to set B = {apple; pear; nut}. \latex{ A } possible one-to-one correspondence is as follows (Figure 12):
\latex{1} → apple; \latex{2} → pear; \latex{3} → nut.
Figure 12\latex{ 3 }\latex{ 1 }\latex{ 2 }\latex{ B }\latex{ U }nut\latex{ A }applepearExample 5
Is set N = {natural numbers} equivalent to set P = {even natural numbers}?
Solution
\latex{ N } is equivalent to \latex{ P }, the one-to-one correspondence is as follows (Figure 13):
\latex{n\longrightarrow 2n \left(n\in \N\right) }
The correspondence is indeed one-to-one:
– to every natural number belongs one even number, its double;
– every even number belongs to one natural number, to its half.
– to every natural number belongs one even number, its double;
– every even number belongs to one natural number, to its half.
Figure 13\latex{ 8 }\latex{ 2 }\latex{ 4 }\latex{ P }\latex{ U }\latex{ N }\latex{ 0 }\latex{ 6 }\latex{ 0 }\latex{ 1 }\latex{ 2 }\latex{ 3 }\latex{ 4 }⯁ ⯁ ⯁
If two sets are equivalent then we say that their orders (cardinal numbers) are equal.
DEFINITION: A set is an infinite set if it is equivalent to one of its real subsets.
For example the set of natural numbers is infinite, because it is equivalent to the set of the even natural numbers.
DEFINITION: A set is a finite set if it is not infinite.
The order of a finite set can be given with a natural number.
For example the set of signs necessary to write down the Roman numerals is finite: <br>R = {M; D; C; L; X; I; V} has an order of \latex{7}.
Note: The infinite sets are like finite sets in many aspects but they also act differently in many other aspects. It can also be seen from the tale below.
Let us imagine a hotel with infinitely many rooms which are numbered by \latex{1, 2, 3, …} There is someone staying in every room so the NO VACANCY sign was placed on the door. But after that a very important new guest asked for a room and the director did not want to send him away in any way. Fortunately the mathematician of the hotel solved the problem. With the help of the loudspeaker he asked all guests to move to a room with a number one greater than their previous room number. So everyone had a new room and the important guest could move into the room with number \latex{1}.
A few weeks later there was a fire in the nearby hotel which was very similar to this hotel and was also full of guests; these guests also asked for accommodation in the first hotel. The mathematician of the hotel solved the problem successfully again and could give accommodation to the new guests. How could he do it?
Exercises
{{exercise_number}}. List the elements of the following sets:
- the set of months with \latex{31} days;
- the set of months with \latex{32} days;
- the set of months the name of which contains the letter \latex{r};
- the set of the days of the week the name of which contains the letter \latex{e};
- the set of the five largest cities of England (based on the number of residents).
{{exercise_number}}. List the elements of the following sets:
- {\latex{x\mid x } is a vowel in the English alphabet};
- {\latex{x\mid x } is the largest river of England};
- {\latex{x\mid x } is a continent};
- {\latex{x\mid x } + \latex{8} = \latex{88}};
- {\latex{x\mid x } + \latex{1} = \latex{x}}.
{{exercise_number}}. Which of the following statements are true and which are false?
- \latex{\left\{\varnothing \right\} } is a finite set
- \latex{\left\{\varnothing \right\} } is an empty set
- \latex{\left\{m;r;p;t\right\} =\left\{p;r;t;m\right\}}
- \latex{4\notin \left\{1;2;3;4;5;6\right\} }
- Bristol \latex{\in} {cities of England}
f) { \latex{x\mid x} \latex{\in \Z} and \latex{x + 1} \latex{\geq x} } is a finite set
{{exercise_number}}. For which of the following sets is it true that \latex{A \subseteq B }?
- \latex{ A } = {3; 4; 5; 6} and B = {3; 4; 5; 6; 7; 8}
- A = {1; 2; 3; 4; 5; 6; 7; 8} and B = {1; 2; 3; 4; 5; 6; 7; 8}
- A = {\latex{x} \latex{\mid} \latex{x} + 18 = 24} and B = {3; 4; 5; 6; 7; 8}
- A = {\latex{x} \latex{\mid} 4 \latex{\leq } \latex{x} \latex{\leq } 7 and \latex{x} is an integer} and B = {3; 4; 5; 6; 7; 8}
- A = {\latex{x} \latex{\mid} 4 \latex{\leq } \latex{x} \latex{\leq } 7} and B = {3; 4; 5; 6; 7; 8}
{{exercise_number}}. List all the subsets of the following sets.
- {\latex{3}; \latex{5}}
- {\latex{a; b; c; d}}
- {⬤;◼;▲}
- {♣; ♦; ♥; ♠}
{{exercise_number}}. If set A = {0; 1}, which of the following statements are true and which are false?
- \latex{1 \subset A}
- \latex{1 \in A}
- \latex{\left\{0\right\} \subset A}
- \latex{\varnothing \subset A}
- \latex{0 \in \varnothing}
- \latex{\varnothing \in A}
{{exercise_number}}. Let the distance between two points A and B be \latex{4} cm. Draw the set of points in the plane which are
- \latex{2} cm away from point A and \latex{3} cm away from point B;
- at most \latex{2} cm away from point A and at most \latex{3} cm away from point B;
- at most \latex{2} cm away from point A or at most \latex{3} cm away from point B;
- at least \latex{2} cm away from point A and at most \latex{3} cm away from point B;
- at most \latex{2} cm away from point A and at least \latex{3} cm away from point B.
{{exercise_number}}. Frank has a: \latex{5} pence, \latex{10} pence, \latex{20} pence, \latex{50} pence and \latex{1} pound coin each in his pocket. How many different amounts can he pay with these coins at the news stall? What is the largest amount he can pay?
{{exercise_number}}. Which of the following statements are true and which are false?
- \latex{\mathbb{N}\subset\mathbb{Q}}
- \latex{\mathbb{R}\subset\mathbb{I}}
- \latex{\mathbb{N}\subset\mathbb{R}}
- \latex{\mathbb{Z}\subset\mathbb{Q}}
- \latex{\mathbb{N}\subset\mathbb{Q}\subset\mathbb{R}}
- \latex{\mathbb{I}\subset\mathbb{Z}}
