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Sets. Operations on sets
"A set is a lot that we think of as a single" founder of set theory - Georg Cantor (1845-1918) - German mathematician, logician, theologian, creator of the theory of infinite sets, which had a decisive influence on the development of mathematical sciences at the turn of the 19th and 20th centuries.
Examples of sets from the surrounding world For example, the set of days of the week consists of the elements: Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday. Many months - from the elements: January, February, March, April, May, June, July, August, September, October, November, December.
Examples of sets in mathematics are: a) the set of all natural numbers N, b) the set of all integers Z (positive, negative and zero), c) the set of all rational numbers Q, d) the set of all real numbers R The set of arithmetic operations - from elements: addition, subtraction, multiplication, division.
Examples of sets in geometry are: a) a set of types of triangles, b) a set of polygons
The intersection of two sets A and B is the set C \u003d A B, which consists of all elements x that lie simultaneously in the set A and in the set B. A B \u003d (x), where x A and x B M \u003d a c
A TASK 1 TASK 2
The union of two sets A and B is the set A B, which consists of all elements belonging to A or B. C \u003d A B \u003d (x), where x A or x B. A - girls of the class, B - boys of the class, C - the whole class
Subset Empty set Equal sets A = B
A=(0,1,2,3,4,5,6,7,8,9) No. 1 What set is given by listing these elements? #2 Set a lot of crocodiles flying in the sky. Sets A = (3, 5, 0, 11, 12, 19), B = (2, 4, 8, 12, 18.0) are given. Find the sets AU B, A B No. 3 B \u003d (A, E, I, O, U, E, Yu, Z)
Solution The fourth pencil case should contain items that are already found in the first three pencil cases, but only once. This is a blue pen, an orange pencil and a red eraser. Answer Blue pen, orange pencil, red eraser. Problem The first pencil case contains a purple pen, a green pencil and a red eraser; in the second - a blue pen, a green pencil and a yellow eraser; in the third, a purple pen, an orange pencil, and a yellow eraser. The content of these pencil cases is characterized by the following regularity: in every two of them exactly one pair of objects matches both in color and purpose. What should be in the fourth pencil case for this pattern to be preserved? Hint Think about whether the fourth pencil case can contain a purple pen.
№ 5 Use Euler circles to draw the intersection of sets K and L if: a) K L b) L K c) K = L d) K L = K K = L L K L K
Solution: Denote by x the number of people who are mathematicians and philosophers at the same time. Then the number of mathematicians is 7 x and the number of philosophers is 9 x . If x 0, then there are more philosophers. What does it mean that x = 0? This means that neither one nor the other exists at all, that is, they are "equal". This is the correct answer, formally satisfying the condition of the problem. And those who pointed it out are doubly well done! Although the solution was also counted by those who analyzed only the case when mathematicians still exist. Answer: If there is at least one philosopher or mathematician, then there are more philosophers. Problem Among mathematicians every seventh is a philosopher, and among philosophers every ninth is a mathematician. Who is more: philosophers or mathematicians? Hint Consider people who are mathematicians and philosophers at the same time.
Sets are usually denoted by large
letters: A, B, X N ,…, and their elements are
corresponding small letters: a,b,x,n…
In particular, the following notation is adopted:
ℕ is the set of natural numbers;
ℤ is the set of integers;
ℚ is the set of rational numbers;
ℝ is the set of real numbers (numerical
straight).
is the set of complex numbers. And right
following:
N Z Q R C
in small letters, and the sets themselves in capital letters.
Affiliation
element
m
many
M
denoted as follows: m M, where the sign is
stylization of the first letter of a Greek word
(is, be)
sign of non-belonging: Sets can be finite, infinite, and
empty.
A set containing a finite number of elements,
called final.
If the set contains no elements, then
it is called empty and is denoted by Ø.
For example:
the set of 1st year students is a finite set;
many stars in the universe - infinite
a bunch of;
a bunch of
students,
Fine
knowing
three
foreign
language
(Japanese,
Chinese
And
French), apparently the empty set.
Ways of specifying sets
There are three ways to define sets:1) set description
Examples: Y=(yΙ1≤y ≤10) – set of y values from
segment
X=(xIx>2) is the set of all numbers x greater than 2.
2) set enumeration
Examples:
A \u003d (a, b, c) - three initial letters of Russian
alphabet
N=(1,2,3…)-natural numbers
3) graphic assignment of sets occurs with
using Euler-Venn diagrams Two sets are given:
And
If there are few elements of the sets, then
they can be indicated explicitly on the diagram. Set A is called a subset of set B.
(denoted A B) if every element
set A is an element of set B:
see figure 1.1
Rice. 1.1
In this case, we say that B contains A, or B covers A
Non-inclusion of set C in set B,
denoted like this: Sets A and B are equal (A=B) if and only
when, A B and B A, i.e. elements of sets
A and B match.
Example: A=(1,2,3), B=(3,2,1), C=(1,2,3,3) are equal.
The set C is the set A, only in it
element 3 is written twice.
Example: A=(1,2), B=(1,2,3)- NOT EQUAL
A family of sets is a set
whose elements are themselves sets.
Example: A \u003d ((Ø), (1,2), (3,4,5)) - a family consisting
from three sets.
Each non-empty subset А≠ Ø has
at least two distinct subsets: itself
set A and Ø. A bunch of
A
called
own
a subset of the set B, if A is B, and B is A.
Designated as follows: A B.
For example,
It is customary to assume that the empty set is
subset of any set.
The cardinality of a finite set M is the number
its elements. Designated M
For example, B=6. A=3.
Operations on sets
Union (sum) of sets A and B(denoted by A B) is called the set C of those
elements, each of which belongs though
to one of the sets A or B. There are three possible
case:
1) A=B;
2) sets have common elements;
3) sets do not have common elements.
Examples:
1) A \u003d (1,2,3), B \u003d (1,2,3), then A B \u003d (1,2,3).
A B=(1,2,3,4,5,6)
3) A=(1,2,3), B=(4,6,8), then A B=(1,2,3,4,6,8) The cases considered
illustrated in the figure
A, B
A
IN
A
IN The intersection of sets A and B
is called the new set C,
which consists only of elements
owned at the same time
sets A, B
Designation C=A B
Three cases are possible:
1) A=B
2) sets have common elements
3) sets have no common
elements. Examples:
1) A \u003d (1,2,3), B \u003d (1,2,3), then A B \u003d
{1,2,3}.
2) A=(1,2,3), B=(2,3,4,5,6), then
A B \u003d (2.3)
3) A=(1,2,3), B=(4,6,8), then A B= The difference of sets A and B is called
set C, consisting of elements
belonging only to the set A and
not belonging to V.
Designation: C=A\B Given two sets:
A=(1,2,3,b,c,d),B=(2,b,d,3).
Then:
A B=(1,2,3,b,c,d)
B subset A
A/B=(1,c)
A B=(2,3,b,d) Properties:
1. Commutativity of union А B=B A
2. Commutativity of the intersection A B=B A
3. Combination law A (B C)=B (A C)
4. The same for the intersection.
5. Distributive relative to the intersection
A (B C) = A B A C
6. Distributive regarding association
A (B C) = (A B) (A C)
7. The law of absorption A (A B) \u003d A
8. Absorption law A (A B)=A
9. A A=A
10. A A=A The Cartesian (direct) product of A and B is
a new set C, consisting of ordered
pairs in which the first element of the pair is taken from
set A, and the second from B.
A=(1,2,3)
B=(4.5)
C \u003d A B \u003d ((1.4); (1.5); (2.4); (2.5); (3.4); (3.5))
The power of the Cartesian product is
the product of the powers of sets A and B:
A B = A ∙ B A B ≠ B A, except if A=B (in which case
equality holds)
Given:
X.x coordinate numeric axis (-,+).
Coordinate numerical axis Y.y (- ,+).
D=X Y
Cartesian product of two axes - point
on surface.
Consider the Cartesian product,
which has the property
commutativity. A=(Ivanov, Petrov)
B = (tall, thin, strong)
A B \u003d Ivanov is tall, Ivanov is thin,
Ivanov is strong, Petrov is tall, Petrov
thin, Petrov strong
slide 2
Natural numbers and actions on them Divisibility. Prime and Composite Numbers Greatest Common Divisor and Least Common Multiple Problems Concept of Set, Intersection and Union of Sets Monomials and Polynomials Factoring a Polynomial Abbreviated Multiplication Formulas Think and Solve Assignments Authors
slide 3
Natural numbers in ascending order can be written as a sequence 1, 2, 3, 4, ... The set of all natural numbers is denoted by N. For natural numbers, arithmetic operations are defined (addition, subtraction, multiplication and division), exponentiation (number a degree n, and n is the result of multiplying the number a by itself n times), the inverse operation to raising to a power is extracting the root (b = ⁿ√a, if a = bⁿ) Addition and multiplication satisfy the commutative law (the law of commutativity): a + b=b+a, a b=b a and the associative law (associativity law): (a+b)+c=a+(b+c), (a b) c=a (b c ), as well as the distributive (distributive) law: (a+b) c=a c+b c natural numbers and actions on them 1 2 3 4 5
slide 4
DIVISIBILITY. SIMPLE AND COMPOSITE NUMBERS. To divide the number a by the number b means to find x, a: b = x such that xb = a. If such a number exists, then we say that a is divisible by b, and the number b is called the divisor of a. Those and only those numbers are divisible by 2 (or 5), the last digit of which expresses a number divisible by 2 (or 5) Those and only numbers are divisible by 4 (or 25), the last two digits of which express a number divisible by 4 (or 25) Those and only those numbers are divisible by 3 (or 9), the sum of the digits of which is divisible by 3 (or 9) Those and only those numbers are divisible by 11, for which the difference between the sum of the digits on the even places, and the sum of the digits in odd places is divisible by 11. A number a other than 1 is called prime if only one and the number a itself are divisors. A number a that has other divisors is called a composite number. Any composite number can be represented as a product of prime numbers, for example: 12 = 2 2 3 = 2² 3.
slide 5
GCD and LCM Among the common divisors of the number and b, you can choose the greatest common divisor GCD (a ; b). For example, gcd (45; 60) = 15. If gcd (a; b) =1, then the numbers a and b are called coprime. Any common divisor of arbitrary numbers a and b divides the greatest common divisor of these numbers. A number divisible by a and b is called a common multiple of a and b. Among the common multiples of a and b, you can choose the least common multiple of the LCM (a ; b). For example, LCM (4 ; 6) = 12. Any common multiple of arbitrary numbers a and b is divisible by LCM (a ; b). Numbers a and b are relatively prime if and only if LCM (a ; b) = a · b.
slide 6
Find the GCD of two numbers: 1. 45 ; 135 2.84; 168 3.5 ; 60 Find the LCM of two numbers: 1. 4 ; 5 2. 6 ; 7 3. 7 ; 8. tasks
Slide 7
The concept of a set One of the fundamental concepts of mathematics is the concept of a set. A set can be imagined as a set (collection) of some objects, united according to some attribute. Set is an indefinable concept. A set may consist of numbers, objects, etc. Each number (object) included in the set is called an element of the set. - this is the set of points 3. The fact that the element a belongs to the set A is written as a € A. for the set of single-valued numbers: A \u003d (0; 1; 2; 3; 4; 5; 6; 7; 8; 9 ) the number 4 belongs to A, but the number 20 does not belong to A
Slide 8
Continuation 4. A set that does not contain elements is called empty and is denoted by the symbol Ø. 5. If each element of one set A is an element of another set B, then they say that set A is a subset of set B. This is expressed by writing A with B. 6. The intersection of sets A and B is a set consisting of elements that belong to each of data sets (Fig. 1) А В С Fig. 1
Slide 9
7. A union of sets A and B is a set consisting of all elements of sets A and B and only of them. The union of sets is denoted by the symbol ں and written C = A ں B = ( x | x € A or x € B (Fig. 2) A B Question: what set is the union of these sets? A = (1 ; 2 ; 5 ; 7) , B = (3 ; 5 ; 7 ; 8) 2. Н = (4 ; 7 ; 67 ; 34 ; 5 ; 2 ), M = (7 ; 89 ; 34 ) 3. K = ( 78 ; 89 ; 56 ; 90), P = (87 ; 98 ; 65 ; 9)
Slide 10
monomials and polynomials An expression that is a product of numbers, variables and natural powers is called a monomial. 2. The degree of a monomial is the sum of the exponents of the variables. For example, 8x³y² is a fifth degree monomial. Monomials that differ only by a numerical coefficient or are equal to each other are called similar. 3. The algebraic sum of monomials is called a polynomial. The degree of a polynomial is the greatest degree of a monomial included in this polynomial. For example, 1+ 2x² - 5x²y³ is a fifth degree polynomial. 4. When taking the sum of polynomials, it is necessary to bring similar terms (terms). To do this, it is enough to add their coefficients and multiply the resulting number by a literal expression.
slide 11
5. When taking the difference of polynomials, it is necessary to take the subtrahend polynomial in brackets, then open the brackets, changing the sign of each term to the opposite, and then bring similar terms. For example, (4x² - 3x + 3) - (3x² - x + 2) = = 4x² - 3x + 3 - 3x² + x - 2 = x² - 2x + 1. 6. To multiply a polynomial by a monomial, it is enough to multiply each member of the polynomial into a monomial and add the resulting products. Division of a polynomial by a monomial product according to a similar rule. 7. To multiply a polynomial by a polynomial, it is enough to multiply each term of the first polynomial by each term of the second and add the resulting products. For example, 5x (x - y) + (2x + y) (x - y) \u003d \u003d 5x² - 5xy + 2x² + xy - 2xy - y² \u003d 7x² - 6xy - y²
slide 12
Decomposition of a polynomial factors When the common factor is taken out of brackets, the expression in brackets is obtained by dividing each term of the polynomial by the common factor. For example, 3ax³ - 6a²x + 12ax² = 3ax(x² - 2a + 12x) Solve for yourself: 1. ab + 2a - 3b - 6 2. 3(x - 2y)² - 3x + 6y
slide 13
abbreviated multiplication formulas a 2 - b 2 \u003d (a - c) (a + c) (a + c) 2 \u003d a 2 + 2ab + c 2 (a - b) 2 \u003d a 2 - 2ab + b 2
