Set Theory Rules are as follows

(A∪A) = A,  (A∩A) = A (idempotent law)
(A∪B)’ = A’∩B’, (A ∩ B) ′ = A ′ ∪ B ′ (De Morgan’s law)
A∪ B = B ∪ A, A∩B = B∩A (commutative law)
(A ∩ B) ∩ C = A ∩ (B ∩ C), (A ∪ B) ∪ C = A ∪ (B ∪ C) (associative law)
A ∪ φ = A, A ∩ U = A, A ∪ U = U, A ∩ φ = φ (identity law)
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C), A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)   (distributive law)
(A’)’ = A (involution law)

Basic Set Theory Properties are:
A st is inherently unordered. This means that the order of elements in a st does not make a new st. Change in order of elements inside a st is feasible. For example if st A = {a, x, c} then A = {a, c, x} = {x, c, a} = {x, a, c} = {c, a, x} = {c, x, a}.
Each element in a st is distinct. Multiple repetition of an element makes no difference. For example if st A = {2, 3} then A = {2, 2, 3, 3} = {2, 3, 3} = {2, 2, 3} = {2, 2, 2, 3, 3, 3, 3} = {2, 3} and so on.
Above rules and properties of Set Theory Help you to solve Set Theory Questions easily.
Let us see some Set Theory Problems and Solutions now:

Q.1) prove that (A ′ ∩ B) ′ ∩ (A ∪ B) = A
Sol.) lets begin with LHS of the above problem:
LHS = ((A ′) ′ ∪ B ′) ∩ (A ∪ B) (by De Morgan’s law)
= (A∪ B’) ∩ (A ∪ B) (by involution law)
= A∪ (B∩ B’) (distributive law/property)
= A ∪ φ (complement law)
= A (identity law) = RHS
Hence, proved.

Q.2) in a community if 70 % people can speak English and 60% people can speak French then find the number of people who can speak both languages.
Sol.) Number of people who can speak English = n(E) = 70 %
Number of people who can speak French = n(F) = 60%
Number of people who can speak both languages will be n(E∩F) = ?
Total number of people in the community n (E∪F)= 100%
Now n (E∪F)= n(E) + n(F) - n(E∩F)
100=70 + 60 - n(E∩F)
100=130 - n(E∩F)
n(E∩F)=30%

Articles – a, an, the

An article is a major concept of language learning. In English an article is considered as an adjective that provides some valuable information about a noun in a sentence. In many other different languages an article is considered as a part of speech. An article can be classified based on definite and indefinite characteristics. These are called definite article and indefinite article.

Definite Article

A definite article is that which defines that a noun is particular and has something definitely addressed. Definite article is used for nouns that have been already used in a speech or for nouns that are uniquely specified. In the English language, there is only one definite article ‘the’. Definite article ‘the’ is used for both singulars and plural nouns. For example: Give me a Barbie doll. Give me the Barbie doll. In the second sentence, a specific Barbie doll is referred to while in the first sentence not any particular Barbie doll is referred.
1) Ram said to his wife, “Please pack the gifts for kids that I bought yesterday”. (Here Ram is speaking about those particular gifts for kids that he bought the previous day.)
2) The Taj Mahal is made by Mughal Emperor Shah Jahan. (Here, Taj Mahal is a uniquely specified noun.)

Indefinite Article

An indefinite article is used for a noun that is not particular and uniquely specified. It can be a noun that the speaker is mentioning for the first time. For example: Give me a Chota Bheem DVD. Give me the Chota Bheem DVD. Here, in the first sentence the speaker is talking about no particular Chota Bheem DVD while in the second sentence the reference is definite. In the English language, there are two indefinite articles ‘a’ and ‘an’. The indefinite article ‘an’ is used before words that begin with a vowel sound. At times even a word that starts with a consonant but sounds like a vowel. For example: an hour, an MBA, etc. Also the indefinite article ‘a’ is used before words that begin with a consonant sound even if the word begins with a vowel. For example: a European, etc. These were the basics about article in English language.

Introduction for Solving Number Patterns

Solving number patterns is the main topic under the solving number systems. In this lesson we have to study about the concepts of divisors, factors, multiples and prime numbers. Also in this topic we have to discuss about properties of number patterns. Here we have to discuss about Divisors, Factors ,Multiples,Even,Odd numbers,Prime numbers and Composite numbers. These are the main topics in the solving number patterns.

Solving Divisors, Factors and Multiples Number Patterns:

Divisors:

Step 1: Consider the number 96. Divide 96 by 4. When we divide 96 by 4, the remainder is 0.

Step 2: 4 is the divisor of 96. Similarly if we divide 96 by 2, we will get 0 as the remainder.

Step 3: Again 2 is the divisor 96. By applying the same process, 1,2,3,4,6,8,12,16,24,32,48,96 are also divisors.

Step 4: Thus the divisors of 96 are 1,2,3,4,6,8,12,16,24,32,48,96.

Note:

The divisor in the division algorithm is different from this divisor. In division algorithm the divisor indicates the denominator only.

Example:

Find the divisors of (a) 28, (b) 35

Solution:

(a) Solving The divisors of 28 are 1, 2, 4,7,14, and 28

(b) Solving The divisors of 35 are 1, 5, 7 and 35

Solving even and Odd Number Patterns and Prime and Composite Patterns:

The numbers that are divisible by 2 is called as even numbers.
If we divide the even number by 2, the remainder is always zero.

The numbers that are not divisible by 2 is called as odd number.
If we divide an odd number by 2, the remainder is always 1

Now observe the following:

(a) 2 + 2 = 4 ; 2 + 6 = 8 ; 4 + 8 = 12 ; 10 + 20 = 30

(b) 2 × 2 = 4 ; 2 × 6 = 12 ; 4 × 8 = 32 ; 10 × 20 = 200

From the above examples the following are true.

Even number + even number = even number
Even number × even number = even number
Odd number + odd number = even number
Odd number × odd number = odd number
These are the main rules used for solving number patterns.

Prime numbers and Composite number patterns

From the above, find the numbers that have only 2 divisors?

2, 3, 5 and 7 have only two divisors.

What are the numbers which have more than 2 divisors?

4, 6, 8, 9 and 10 have more than 2 divisors.

The natural numbers that have only two divisors are known as the prime numbers.
The natural numbers that have more than two divisors are known as the composite numbers.

Example:

Solving prime numbers between 20 and 30.

Solution:

The numbers between 20 and 30 are

21 22 23 24 25 26 27 28 29

∴ the prime numbers are 23 and 29

Definition of parabola

Definition of parabola:
A curve in a plane having two ends and one vertex is called a parabola. It is a conic section. That means it is a section of a double cone intercepted by a plane. When a double cone is cut by a plane such that the plane is parallel to any one slant length of the cone, the cross section thus obtained is called as such.

However in co-ordinate geometry we do not study parabola problems as sections of double cone. We use the characteristic property of these to define it and use that to find its equation.

It so happens that in the plane of every parabola there is a line (called the directrix) and a point (called the focus) such that for every point on this the distance from the directrix and the distance from the focus are found to be equal. This is the definition most commonly used in co-ordinate geometry to define these and to derive its equation.

Types of parabola:
Primarily these are of two types and they are horizontal and vertical. In simple terms, the horizontal form of this is x as a function of y. Therefore the equation of a horizontal parabola would be x = f(y) type. On the other hand in a vertical form of this the equation is y as a function of x. Therefore it would look like this: y = f(x).

Another way of classifying these would be: positive and negative parabola. For the horizontal form, if it opens towards the positive side of the x axis, it is called the positive para-bola; where as if it opens on the negative side of the x axis, it is called the negative para-bola. Similarly, for the vertical form, if it opens towards the positive y axis it is called a positive para-bola and if it opens towards the negative side of the y axis, it is called a negative form of same. Below are the pictures of the various types of these:

1. Horizontal positive form of the same:

As you can see in the above picture, this one opens to the right.
X = f(y) = p(y-k)^2 + h
2. Horizontal negative form:

The above form opens to the left.
X = f(y) = -p(y-k)^2 + h

3. Vertical positive form of the same:

A vertical positive form opens up (as can be seen in the picture above).
Y = f(x) = a(x-h)^2 + k

4. Vertical negative form:

Lastly the vertical negative form would open down.
Y = f(x) = -a(x-h)^2 + k

How to calculate Area of Ellipse?

An ellipse is nothing but the planar curve which is resulted by the intersection by a plane of a cone in a way such that it will produce a closed curve. Circles can be said as a special case of ellipse which can be obtained by cutting in orthogonal plane of the cone’s axis. Ellipse can also be said as the locus of all the points in the plane in which the sum of the distances of two fixed points will be a constant. Before going into the formula for area of an ellipse, it is essential to know the elements of ellipse which forms the structure of it. An ellipse is said to be a smooth and closed curve symmetric about its vertical and horizontal axis. The longest diameter of an ellipse is called as major axis and the shortest diameter of an ellipse is called as the minor axis. These both axes are the lines that pass through the centre of an ellipse.

With the measurement of both these axis only, the area of a ellipse will vary. Each of the above said axes bisects perpendicularly with the other. Also, the sum of the distance from the two focus of the ellipse to any point P on the ellipse will be equal to the major axis. It can also be said that each axis cut the other equally into two parts and they cross at right angles to each other. Also, if these both axes are equal in length, then it is called as circle.

Area of an Ellipse Formula
The standard equation for representing the ellipse equation is given as,
X^2 / a^2 + Y^2 /b^2 = 1.

This equation is a standard equation in which when the ellipse is centred with origin.
But algebraically the ellipse area formula is given in other terms with the help of semi major axis (half of major axis) and semi minor axis (half of minor axis). Thus the formula for area of ellipse is given as Pi*a*b, where ‘a’ is the semi major axis and ‘b’ is the semi minor axis. This formula is actually arrived from the formula of circle Pi multiplied by radius square. Here the radius is split into semi major and semi minor axis.
Also at some special cases, when an ellipse is given by the implicit equation which is given as,
AX2 + Bxy + CY2 = 1, then the area of the ellipse would be 2*Pi whole divided by square root of 4AC subtracted with B square.

Understanding Fractions Better

Numbers have always fascinated humankind. Indians are credited with the invention of zero. Indians have always been good with numbers and the world looks at us with awe. We have great strides and come a long way. From natural numbers, whole numbers, integers, real numbers and complex numbers. We have also studied fractions. Let us what are fractions to understand the concepts of denominator numerator. A number can written in the form of A/B where A is an integer and B is an integer. A can be called the numerator and B can be called the denominator. So we see numerator v/s denominator to extend the concept further. The numerator is written on top of the horizontal line and the denominator is written below the horizontal line. So we understand what is a numerator and denominator if we didn’t know it earlier.

We must understand numerators and denominators to understand fractions better. Operations of addition, subtraction, multiplication and division between the fractions can be performed. Here both numerator and also the denominator come into picture. If the denominators are dissimilar they must be made similar to proceed further with the operations. For this we need to take the LCM of the denominators. Taking the LCM is a simple operation. Unless the denominators are equal we cannot proceed further. Once the denominators are equal, we can just add the numerators if the operation to be performed is addition. If subtraction is to be performed we just need to subtract the numerators after making the denominators equal. So we must know the definition of numerator and denominator to perform these operations. Only when we know it, we can do something.

After making the denominators equal if they are not, we proceed to perform addition, subtraction or any operation with fractions. Sometimes the fractions can be in the form of mixed fractions. We need to be careful in such cases. We must first convert them into improper fractions without proceeding. They cannot be directly used. For the process of finding LCM we need improper fractions in our hand. Once we get the improper fractions we first check whether their denominators are same or not. If they are same, our problem becomes very simple and easy. The numerators can be just added or subtracted as the case may be. If they are not equal only then the process of taking LCM is to be performed and addition or subtraction is to be done.

Distributive Property of Addition

The distributive property is expressed in the mathematics expressions as following equation: a (b + c) = ab + ac. You can understand this as the sum of a (b + c) is corresponding to the sum of a times b and a times c. Distributive property of addition would be inaccurate to multiply ab and just add c, or to multiply ac and add b.

Order of Operations:

The distributive property of addition that remind that all contained by the parenthesis needs to be multiplied by the outside number. Distributive property, when they are knowledge the order of operations.

Concept of that the problems anywhere present are different mathematical operations, such as multiple, addition, subtraction, parenthesis, you have to work in a certain order to get the right answer. This arrange is the parenthesis, exponent, multiplication,division , addition and subtraction, that may be abbreviate to the  PEMDAS.

Example Problems (distributive Property of Addition):

Example problems:

When you include a mathematics problem that use parenthesis you need to solve what’s in the parenthesis first, by you can move about on to solve further problems. If the mathematics problem largely has known numbers, it is somewhat easy to solve. 2(10+5) becomes 2(15) or is also equal under the distributive property of addition is 2(10) + 2(5). What obtain additional difficult is when you are functioning with variables (such as a, b, x, y, and so on) in algebra, and when these variables cannot be joint together.

Consider the equation 5(12a + 2), but we don’t know what the variable a stand for, we can’t add 12a + 2, but using the distributive property still allow us to just this expression because we identify this equation is equal to 5(12a) + 5(2). In order to simply the expression we can take each part separately and multiply it to 5, and we get 60a + 10.