Skip to main content

SETS, RELATIONS & FUNCTIONS

SETS, RELATIONS & FUNCTIONS

Sets are a fundamental part of mathematics and it’s knowledge is very important in present. It defines the concepts of relation and functions. These functions are also very essential for present day maths.

SETS

Sets are denoted by capital letters. They are similar to matrices whose elements are denoted by small letters. But in sets an element comes only at once. Sets are matrices which give a sense of belonging. This sense of belongingness is represented by   .We can write anything in a set like rivers of India, Wonders of the World .etc. 
For an example let there be a set of positive single digit. Let the set be denoted as A
Therefore, 8 A
 Also we can write, 17 A
Set can be represented by two types 

ROSTER

In roster form we write elements in curly brackets and separate them with comma. 
If there be set of positive integers less than 10. Therefore the set will be {1,2,3,4,5,6,7,8,9} 

SET- BUILDER FORM

In set-builder form we take a variable x and then gives its condition. For eg: we can write the above example as
{x: x is positive integer less than 7}
There are some standard symbols for specific sets like
N : set of all natural numbers
Z : set of all integers numbers
R : set of all real numbers
Z+/Z- : set of all positive/negative integers
Q+/Q- : set of all positive/negative rational numbers
R+/R- : set of all positive/negative real numbers

TYPES OF SETS 

There are various kind of sets. 
EMPTY SET
Set having no element is called an empty set
 For eg: {x: x < 0, x is a whole number }
Whole number is 0 and more than 0. Thus it is a empty set
FINITE AND INFINTE SETS
Sets which have limited elements are called finite set 
For ex: {x: 0 < x < 15}
Sets which have infinite number of sets are called infinite set 
For ex: {x: x R}
EQUAL SETS
Sets which have same elements are called equal sets. The elements can be in any order
For ex: {2, 5, 6, 7} and {5, 7, 6, 2} are equal sets
             {x: 0<x<11} and {x: x is an natural number less 11}
NOTE : in this articles I can use the following sign
              ≤ - more than or equals to 
              ≥ - less than or equals to
SINGELTON SETS
Sets having only single element are called singleton sets. 
For ex: {x: 1<x<3, x Z} is an singleton set.
SUBSET
Subset means when every element of a set A is in another set B. In this case we write … 
A B
If a A and A B then a B
if A is not a subset of B then we write
A B
For ex: A = {1, 3, 5} , B = {2, 3, 1, 6, 5, 7} then A B
If A B and B A then the two sets are equal
Remember A is a subset of B is not vice versa, which means that if A is a subset of B then it is not necessary that B is a subset of A.
Also, 
  • Every set is a subset of itself
  • Empty set is a subset of every set
  • The maximum number of subsets of a set having n elements is 2n

INTERVAL AS SUBSETS OF SETS OF REAL NUMBERS   
Let a, b R. Let a set is in the form
{y: a<y<b}
Then this is known as an open set which means that ‘a’ and ‘b’ are not included in the set. Such sets can be written as (a, b) with curly brackets.
If ‘a’ and ‘b’ are included in the set then we write [a, b] with square brackets
There can be set known as the semi-open set. In this set it is open at one end and closed at the other. Like … [a, b)…closing at ‘a’ and opening at ‘b’ or …(a, b]… opening at ‘a’ and closing at ‘b’
This can be represented in a number line with the help of filled and unfilled dots. Filled dots for closed and unfilled for opened 

POWER SET OF A SET

Power set is a set containing all the subsets of a set A. It is denoted by P(A). In P(A) all elements are sets. If A has n elements then P(A) has 2n elements
If A = {a, b} then P(A) = {φ, {a}, {b}, {a, b}}

UNIVERSAL SET

Out of given number of set if there is a set such that every other set is a subset of it then it is called a universal set
If there is are sets A, B, C, D
Such that A D, B D, C D then D is the universal set. It is represented by ‘U’

OPERATION OF SETS

UNION OF SETS
Union sets means that a set which contains all the elements of A + all the elements of B. If A and B have some common elements then we do not repeat them.
It can be represented by
A B {x: x A or x B}
The colored region is the desired set. 
INTERSECTION OF SET
Intersection of set is a set which have only the common elements of the two sets
It can be represented by , A B

the yellow region is the intersection of A and B
COMPLEMENT OF A SET
Complement of a set is the difference of the set and its union set
It is denoted by A
     A + A = U



Comments

Post a Comment

Popular Posts

Cathode Ray Experiment

This experiment was conducted by J.J. Thomson (Sir Joseph John Thomson) in the year 1897. This experiment proved that atom is made up of fundamental particles which are much smaller than the smallest atom 'hydrogen' This experiment helped to discover electron. According to J.J. Thomson, the cathode rays consisted of very light, small and negatively charged particles. He named the particles "corpuscles" which were later known as electrons

Android Versions Named After Sweet

Have you ever thought why are Android versions always named after sweet names ?? Everytime a new Android version is launched its name is kept after a sweet name. Many people have researched about this topic and many have asked Google also. Have you ever tried to find out the core reason behind this? If not then you would find the answer here . First of all let us first see what Google says about this : In 2008 i.e. the year when Android was launched a reporter asked the reason for the same. At that time Google said “It’s kind of like an internal team thing, and we prefer to be a little bit — how should I say — a bit inscrutable in the matter, I’ll say,” said Randall Sarafa, a Google spokesman. “The obvious thing is that, yeah, the Android platform releases, they go by dessert names and by alphabetical order for the most part."

Anode Ray Experiment

→Anode ray experiment was conducted by E Goldstein. →These rays are also known as canal rays. →This experiment helped in the discovery of the proton. Apparatus Used A discharge tube  was taken in which there were 2 electrodes i.e. Anode(+ve) and the cathode (-ve). The tube was filled with an inert gas. A perforated or porous cathode was used. A layer of zinc sulphide was placed at the back of the cathode. There was a vacuum pump in the tube. High voltage (5000v-10000v) was allowed to flow through the system. It was observed that when the gas was at 1atm(atmospheric pressure ) no change was seen in the tube.  When the   pressure   was decreased inside the tube, a glow could be seen at the back side of the cathode.

Rutherford Alpha Ray Scattering Experiment

Rutherford Alpha Ray Scattering Experiment Hey, Guys, most of you might have heard about the alpha ray scattering experiment and if you want to know in detail about Rutherford's model and the experiment he conducted, this is the right place for you... But first: Things You Must Know Some basic information that will help you understand rutherford experiment properly: Proton is a sub-atomic particle which is positively charged and has a mass of 1u. Alpha particles are helium atom with a charge of +2 as they have lost 2 electrons. Alpha particles have an atomic mass  of 4u. Gold is highly malleable and can be beaten into very thin sheets. Experiment Rutherford conducted his experiment in the following way: Rutherford took a very thin gold foil and bombarded it with high energy alpha particles. He placed a layer of zinc sulphide on the walls where the experiment was taking place because when alpha particles strike zinc sulphide layer, it results i

Animal and Plant Cells

 Cells Cells are the basic functional, biological and structural unit of life. The word cell is a Latin word meaning ‘small room’. Cells are also known as building blocks of life.  The branch of science that deals with the form, structure, and composition of a cell is known as Cytology. All organisms around us are made up of cells. Bacteria, ameba, paramecium, algae, fungi, plants and animals are made up of cells.  Cells together form tissues. And tissue together makes an organ. History Of Cell The cell was discovered by Robert Hooke in 1665. He assembled a simple microscope and observed a very thin slice of cork under his primitive microscope. The cork was obtained from the outer covering of a tree called bark. Robert Hooke observed many little-partitioned boxes or compartments in the cork slice. These boxes appeared like a honey-comb. He termed these boxes as the cell. He also noticed that one box was separated from another by a wall. What Ho

Sectors of Indian Economy

Hey there all the tenthees. You guys must be facing a lot of pressure as it is a board class as quoted by many teachers. You need to take care of each and every stuff including the small stats too. But unfortunately most of the Economics books we got do not have the graph 3 for the chapter Sectors of Indian Economy. No worries guys you came to the right place. We are providing you with the graph of December 2015 edition which is the latest graph which we need. All the best for your board exams and here you go Once again All the very best guys !!!

Doppler Effect

Doppler Effect states that there is a change in the frequency of sound because of the relative motion between the observer or the listener and the source of the object. In simpler words when the source and the listener are in a relative motion there is a change in the frequency of sound and this effect of a change in frequency is known as the Doppler's Effect of sound. When the objects move apart the frequency decreases. When the objects move towards each other the frequency increases. The change in the frequency of the sound is given by the formula: