Set theory

We deal with many types objects in mathematics and when we combine them and out them together, we call them Sets Math and study of that collection of objects is the theory of sets.

The objects of the sets are usually called elements.

Set Theory Notation– We know that the collection of elements is termed as sets. We usually write a st in curly brackets.
For example: - A st of first five even numbers can be written as {2, 4, 6, 8, and 10} or the saet of first seven natural numbers is written as {1, 2, 3, 4, 5, 6, 7}.

The intersection of two sets is symbolized by inverted U and the union of two sets is written as with symbol U. Union of sets A and B means the saet of all objects that is member of both the sets and this can be written as A U B.

The intersection of two sets that is A intersection B is all the elements that are part of both A and B. If A is contained or part of B than we can call A as subset of B or if B has A contained in it then we can call B as superset of A.

We use different types of axioms in the theory of sets. Axioms can be logical or non-logical. Logical axioms tells us the universal truth which is like a + b = b + a.

An axiom act as a starter to derive the proofs similarly in Set Theory Proofs, we make use of Axiomatic Set Theoryto derive various proofs. 

There are many types of axioms that are used in the axiomatic theory which are axiom of existence, axiom of equality, axiom of pair, axiom of separation, axiom of union, axiom of power, axiom of infinity, axiom of image, axiom of foundation and axiom of choice.

Set Theory Problems are simplified using the axioms of the theory.

We have different types of special sets that we use in mathematics and that are used in the theory of sets.
For example the saet of all integers is denoted by Z where Z = {…-3, -2, -1, 0, 1, 2, 3…} and st of all prime numbers is denoted by P where P = {2, 3, 5, 7, 11, 13, 17, 19, 23…}

LCM (least common multiple)

LCM is an abbreviation for Least Common Multiple’. It is also called as ‘lowest common multiple’ and less popularly as ‘smallest common multiple’. For a given set of two or more integers, the least common multiple is the least integer that can be divided by all the integers in the given set.

The fundamental method of finding the smallest common multiple for a set of integers is first list out the multiples of each integer. Pick up the common integers that appear in all such lists. Now figure out the lowest among them. It is the least common multiple. For example, let us try to find the least common multiple of 4 and 6.
The multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40 …
The multiples of 6 are: 6, 12, 18, 24, 30, 36, 42 …
The integers that are seen common in both the lists are: 12, 24, 36 …
The smallest integer among these commons is 12.
Therefore, the smallest common multiple of 4 and 6 is 12.

There is another method to find the lowest common multiple by using the following formula.
LCM of x, y = (x*y)/(GCF of x and y)
Let us try this method to find the least common multiple of 12 and 16.
The GCF of 12 and 16 is 4.
Therefore, the least common multiple of 12 and 16 = (12*16)/(4) = 3*16 = 48.

In cases where you need to find the lowest common multiples of many numbers, especially large, you can find that by prime factorization. The method here is to do the prime factorization of each number. The lowest common multiple is the product of the highest power of each prime number, out of all factors. Let us find the least common multiple of 9 and 12 by this method.
The prime factorization of 9 is 3*3 = 32.
The prime factorization of 12 is 3*4 =3*2*2 = 3* 22.
The product of the highest powers of prime numbers is 32* 22= 9*4 = 36.
Hence, the smallest common multiple of 9 and 12 is 36.

The knowledge of lowest common multiple for a given set of integers are greatly helpful in finding lowest common denominator for a given set of fractions. The lowest common denominator is the lowest common multiple of the denominators of the given fractions. Therefore, LCM problems are mostly found in such applications. For example the lowest common divisor of the fractions (1/3) and (1/5) is 15, which is the lowest common multiple of 3 and 5.

Word Problems on Multiplication

Word problem is one of the most important concepts in basic arithmetic. A word problem is nothing but a text representation of a mathematical operation such as addition, subtraction, multiplication and division. Word problems make it easier for one to solve a mathematical problem. Let’s have a closer look at word problems on multiplication.

Word problem on multiplication is a textual representation of multiplication operation. For example: There are 18 babies in this day care home. Every baby requires two metal sipper bottles each. How many metal sipper bottles one needs to buy? This is the text representation of multiplication problem: (18*2=?). The answer is 9.

Solving Word Problems on Multiplication
Solving word problems on multiplication includes certain steps. Firstly, the numbers to be multiplied by and with needs to be identified. Secondly, the word problem needs to be converted to a mathematical problem and then finally multiply the number. For example: Grade v has 30 students and each student needs 12 Crayola crayons. How many Crayola crayons I need to buy in total? Identifying the numbers, we get to be multiplied with i.e. 30 and the number to be multiplied by i.e. 12. Converting the word problem to a mathematical operation and then multiplying, (30*12=360). Therefore, the answer is 360 Crayola crayons.

Examples of Word Problems on Multiplication

1. Maria has three children. She wants to buy three pairs of socks for each from online shop baby store. How many pairs of socks she needs to buy from online shop baby store?
Answer: 3 children, 3 pairs of socks each
3*3 = 9 pairs of socks.
Therefore, Maria needs to buy 9 pairs of socks.
2. There are 50 people in the show. How many apples do one needs to get so that each person get at least 5 apples?
Answer: 50 people, 5 apples each
50*5 = 250 apples.
Therefore, the answer is 250 apples.
3. The man has two sisters. Each needs five notebooks. How many notebooks do the man needs to buy in total?
Answer: 2 sisters, 5 notebooks each
2*5 = 10 notebooks.
Therefore, the answer is 10 notebooks.

Adjective and its types

Adjective is one of the eight parts of speech. Adjective is a part of speech that says something about the noun or describes a noun. For example: Disney store India collection has many fun toys for kids. Here, ‘many’ is the adjective that is talking something about the noun. There are different types of adjectives, namely, adjectives of quality, adjectives of quantity, adjectives of number, demonstrative adjectives and interrogative adjectives. Let’s have a closer look at each of these types along with examples in this post.

Adjectives of Quality

These types of adjectives answer to the question “of what kind”. Common examples of adjectives of quality are: beautiful, ugly, big, small and more. For example: Arun bought some very beautiful kids’ toys and accessories from online Disney store India collection.

Adjectives of Quantity

These types of adjectives answer to the question “how much”. Common examples of adjectives of quantity are: some, little, any, enough and more. For example: DK books India collection has some good collection of story books and science books for kids.

Adjectives of Number

These types of adjectives answer to the question “how many”. Common examples of adjectives of number are: one, two, three, four and more. For example: I bought three encyclopedia books from DK books India collection yesterday.

Demonstrative Adjectives

These types of adjectives answer to the question “which type”. Common examples of demonstrative adjectives are this, that, these, those and more. For example: These pen packs from ELC India store is of real good quality.

Interrogative Adjectives

These types of adjectives are used to ask questions about the noun. Common examples of interrogative adjectives are: what, which, whose and more. For example: What did you buy from ELC India collection?

More Examples:

They live in a beautiful house. (Beautiful is an adjective of quality.)
She had enough suffering from him. (Enough is an adjective of quantity.)
I got three educational toys from ELC India collection. (Three is an adjective of number.)
Those apples we bought last Monday are very nice. (Those are demonstrative adjective.)
Which is your purchased toy from online? (Which is the dear one?

These are the basics about adjectives and its types.

Set Theory and Logic

Set Theory and Logic is branch of mathematics which concerns with the study of sets. Sets are the collection of objects or data which are known as elements of the set(s).
 A is said to be a subset of B if each element of A is an element of B. we denote it as: A ⊂ B.
 There are different binary operations applied on two sets as follows:

Union of sets A and B (A∪B) gives all the elements which are either in A or B. eg: if A={4,5,6,7} and B={12,3,4,7} then A∪B is {3,4,5,6,7,12}.

Intersection of sets A and B gives the elements which are common in both sets. Eg:if A={4,5,6,7} and B={12,3,4,7} then A∩B={4,7}.

Difference of sets A and B give the elements of A that are not in B.  Eg:if A={4,5,6,7} and B={12,3,4,7} then A-B={5,6}.

Symmetric difference (A∆B)of sets A and B gives the elements which are member of either of the sets but not of both. Eg:if A={4,5,6,7} and B={12,3,4,7} then {3,5,6,12} is symm. diff.

Set Theory Venn Diagram is a diagram which projects all possible relationships among some sets. Sets  in a diagram are represented by a closed curves, normally circle in a plane. Above operations can also be represented through these diagrams as shown:

(A-B)

Set Theory Formulas 
A∪A=A
A∩A=A
A∪B=B∪A
A∩B= B∩A
(A∪B)∪C=A∪(B∪C)
(A∩B)∩C=A∩(B∩C)
(A∪B)C= AC∩BC
(A∩B)C=AC∪BC
E(A∪B)=E(A)+E(B)–E(A∩B)
E(A∪B∪C)=E(A)+E(B)+E(C)–E(A∩B)–E(A∩C)–E(B∩C)+E(A∩B∩C)

Infinite Set Theory deals with infinite sets. Infinite sets are those which are not finite or the number of elements is infinite. These sets are countable or uncountable. Eg: set of positive integers is countable set while that of real numbers is uncountable.  

Probability Set Theory explains probability of outcome of an experiment. The possible outcomes of an experiment are represented using sets. For example when a coin is tossed its possible outcomes are: head and tail. So outcome is given as: S ={head, tail} or {H, T}. Similarly, if two coins are tossed then possible outcomes are given as: {HH, HT, TH, HH}.
The formulas of sets theory are also applicable here. If A and B are two events then P(A∪B)=P(A)+P(B)-P(A∩B). Here P is probability of an event to occur.

Set Theory Proofs Examples to understand this topic clearly.
Example) Let P, Q and R are three sets. If P ∈ Q and Q ⊂ R, then P ⊂ R is true or false? If not true, give example.
Solution)
False. Let P={3}, Q={{3}, 2} and R={{3}, 2, 4}. Here P∈Q as P={3}
and Q⊂R. But P⊄R as 3∈P and 3∉R.

components of algebra

In this article components of algebra, we will discuss algebra and the components of algebra. Algebra is a part of mathematics, which mainly deals with the expressions, variables and arithmetic operators. Algebra leads to understand the mathematical concepts from the childhood. Generally any part of the mathematical word problems is converted to algebraic equations prior since which is the easiest method to find the solution by solving the algebraic equation. Let us discuss some example for components of algebra.

Components of Algebra:

Consider the following algebraic equations given below,

`x+5(x+8)=70`

The above line is fully named as algebraic equation or algebraic expression

In the above algebraic equation,

x is variable

‘+’ algebraic operator

Numbers are the constant

Consider the following algebraic equations given below,

`x+y=15`

Algebraic Equation may contain the single variable,two variable or it may contain multivariable .Here the above equation contains the two variable.

Consider the following algebraic equations given below,

`x+sin x +e^(x)=6`

Algebraic equation may contain some other functions such as trigonometric function exponential function or logarithmic function.

Example Problem for Components of Algebra:

Example problem 1- Components of algebra

An integer is six more than another integer, sum of the twice the smallest integer and four times the greatest integer is 54. What is the greatest integer?

Solution:

Consider Smallest integer x

Greatest integer x+6

2x+4(x+6) =70

2x+4x+24=70

6x+24=54

6x=30

x=5

Smallest integer x=5

Greatest integer x+6=11

Example problem 2- Components of algebra

The ages of Paarthi and Sabari differ by 12 years when comparing. If 3 years ago, the elder one be 5 times as old as the younger boy, find Paarthi and Sabari present ages.

Solution:

Consider the age of the Paarthi be x years

The age of the Sabari = (x + 12) years

:. 5 (x - 3) = (x + 12 - 3)

5x-15=x+12-3

4x=15+12-3

4x=27-3

4x=24

x=6

Paarthi present age is 6 years

Sabari present age is 18 years

Using integrals to find arc length

Arc length problems:
If we were to find the length of a function that is continuous over the interval [a,b] for the function y = f(x), we would call that a problem of finding arc length.

Arc length as limit of sum:
Since we are trying to estimate the length of a curve, we would do that by splitting the curve into ‘n’ parts. The width of each part would be 𝛥X . Each of the points on the curve would be P0,P1,P3,… Pi,….  Pn. So that P0 is the starting point of the curve at x = a and Pn is the ending point of the curve at x = b.

The length between P0 and P1 can be denoted by |P0 – P1|. The length between P1 and P2 would be |P2-P1| so similarly the length between Pi-1 and Pi would be |P(i-1) – Pi|. In such case the approximate length of the curve would be L = ∑_(i=1)^n▒|P(i-1)- Pi| . But here what is n? As n gets larger and larger, the distance |P(i-1) – Pi| gets smaller and smaller and retrospectively the L becomes more accurate. So now we can say that L = lim┬(n→∞)⁡∑_(i=1)^n▒|P(i-1)- Pi| .

The co-ordinates of any point Pi are (xi,yi) and similarly that of P(i-1) are (X(i-1), Y(i-1)). Therefore the distance between P(i-1) and Pi using distance formula would be:
|P(i-1) – Pi| = √[(X(i-1) – xi)^2 + (Y(i-1) – yi)^2].
Here we already established that the distance between two consecutive points on x axis was 𝛥x and let Y(i-1) – Yi = 𝛥yi. Then,
|P(i-1) – Pi| = √[ 𝛥x^2 + 𝛥yi^2]
So that now,
L = lim┬(n→∞)⁡∑_(i=1)^n▒〖√[ Δx^2 + Δyi^2]〗

Arc length integral:
We know that limit of a sum can be written as an integral. Therefore the above limit of sum can be written as an integral as follows:
L = ∫_a^b▒〖√[ Δx^2 + Δyi^2]〗.
Since we are talking of integrals we can replace the 𝛥x and 𝛥yi with dx and dy, so we have:
L = ∫_a^b▒〖√[ dx^2 + dy^2]〗. Factoring out the dx^2 we have:
L = ∫_a^b▒〖√dx^2[1  + dy^2/dx^2]〗. Bringing the dx^2 outside the root house, makes it dx. So that now we have:
L = ∫_a^b▒〖√[1  +dy^2/dx^2 ]dx〗. That is the same as writing,
L = ∫_a^b▒〖√[1  +(dy^ /dx^ )^2]dx〗
This is the formula to be used to calculate arc length of a curve. Alternatively the formula can also be stated as:
L = ∫_a^b▒〖√[1  +(f^' (x))^2]dx〗