Relations and Functions

Class XII

Relations and Functions

All functions are relations but all relations are not functions.

Relations

A relation R in a set A is a subset of AxA.

Empty relation φ and  universal relation AxA are extreme relations.

Empty Relation

A relation R in a set A is said to be empty, if no element of A is related to any element of A.

R = φ ⊂AxA

Universal Relation

A relation R in a set A is said to be universal relation, if each element of A is related to  every element of A.

R = AxA

Method to Represent a Relation

There are two methods to represent a Relation.

1. Roster Method 2. Set Builder Method.

For Example

Relation R in the set

R = {1,2,3,4}

is a relation represent roster method.

R = {(a,b):b = a +1}

is a relation represent set builder method and read as R is a relation as a related to b or aRb if and only if b = a+1.

Reflexive, Symmetric and Transitive and Equivalence are the types of Relation.

A relation R in set A is said to be reflexive, Symmetric and Transitive if and only if 

1. Reflexive, if (a,a) ε R for every aεA.
2. Symmetric, if  (a,b)εR implies that (b,a)εR for all elements of A.
3. Transitive, if (a,b)εR and (b,c)εR implies (a,c)εR for all elements of A.
4. Any relations which satisfy all of these conditions are said to be Equivalence Relations or other words the relation which is reflexive, symmetric and transitive is said to be Equivalence Relation.

Exercise 1.1

NCERT Text Books

Question No. 1 Determine whether each of the following relations are reflexive, symmetric and transitive.

(i) Relation R in the set A = {1,2,3, ................. 13,14}

defined as 

R = {(x,y):3x - y = 0}

Solution:

Roster form 

R = { (1,3), (2,6),(3,9), (4,12)}
(a,a) is not the element of  R for every elements of  A.
Hence, it is not reflexive.

(a,b)εR does not imply that (b,a)εR for all elements of A.
Hence, it is not symmetric.

 (a,b)εR and (b,c)εR does not imply (a,c)εR for all elements of A.
Hence, it is not transitive.

Therefore, this is none any one of these three.

(ii) Relation R in the set N of natural numbers defined as 
R = {(x,y):y = x + 5 and x<4}

Solution:
Roster form
R = { (1,6),(2,7), (3,8)}

(a,a) is not the element of  R for every elements of  A.
Hence, it is not reflexive.

(a,b)εR does not imply that (b,a)εR for all elements of A.
Hence, it is not symmetric.

 (a,b)εR and (b,c)εR does not imply (a,c)εR for all elements of A.
Hence, it is not transitive.

Therefore, this is none any one of these three.

(iii) A = {1,2,3,4,5,6}
R = {(x,y):y is divisible by x}

Solution:
Roster form
R = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6), (1,2), (1,3), (1,4), (1,6), (2,4), (2,6), (3,6)}

From roster form it is clear that;
(a,a) is the element of R for every element of A.
Hence, it is reflexive.

(a,b)εR does not imply that (b,a)εR for all elements of A.
Hence, it is not symmetric.

 (a,b)εR and (b,c)εR does not imply (a,c)εR for all elements of A.
Hence, it is not transitive.

Similarly we can do remain problems of this section.

Communication Systems

Class XII
Chapter 15

Pointwise Details of This Chapter

Communication Systems

1. J.C. Bose, F.B. Morse, G. Marconi and Alexander Graham Bell have outstanding contribution in the field of communication system.
2. Every communication system has three essential elements 1. Transmitter, 2. Medium/Channel and 3. Receiver.
3. Receiver and Transmitter are connected by Medium or Channel.
4. Medium or Channel is in the form of wires or cables or it may be wireless.
5. Point to Point and Broadcast are two basic modes of transmission.
6. In point to point communication, communication takes place over a link between a single transmitter and a receiver.
7. Telephony is an example of point to point communication.
8. In Broadcast communication, communication takes place between many receiver and a single transmitter.
9. Radio and Television are the examples of Broadcast communication.
10. Transducer is a device which converts one form of energy into another.
11. Electrical transducer is such device which converts pressure, displacement, force, temperature, etcetra into corresponding signals at its output.
12. Information converted in electrical form and suitable for transmission is called signal.
13. Analog and Digital are two types of signals.
14. Analog signals are continuous variation of voltage and current.
15. Sound and Picture signals in television are analog in nature.
16. Signals which can take only discrete stepwise values. Binary system is used in digital electronics employs just two levels of a signal. 0 corresponding to low level and 1 corresponding to high level of voltage and current.
17. There are several coding scheme in digital communication such as Binary Coded Decimal (BCD), American Standard Code for Information and Interchange (ASCII)
18. Nowadays one more signal that is Optical Signal.
19. Unwanted signals that tend to disturb the transmission and processing of message signals in a communication system is said to be Noise.
20. One which process the incoming message signal so as to make it suitable for transmission through a channel and subsequent reception is said to be transmitter.
21. One which extracts the desired message signals from the received signals at the channel output.
22. The loss of strength of a signal while propagating through a medium is said to Attenuation.
23. The process of increasing the amplitude of a signal using an electronic circuit is said to be Amplifier.
24. Range is the largest distance between a source and a destination up to which the signal is received with sufficient strength.
25. Band width refers to the frequency range over which an equipment operates or the portion of the spectrum occupied by the signal.
26. Modulation is process in which original low frequency message or information signal cannot be transmitted to long distances because of reasons the transmitter, information contained in the low frequency message signal is superimposed on a high frequency wave, which acts as a carrier of the information.
27. Demodulation is the reverse process of modulation in which retrieval of information from the carrier wave at the receiver.
28. Repeater is the combination of a receiver and a transmitter which picks up the signal from the transmitter, amplifies and retransmits it to the receiver. Repeater are used to extend the range of a communication system.
29. Frequency range 300 Hz to 3100 Hz is adequate for Speech signal.
30. A bandwidth of 20 kHz is required for musical transmission.
31. A bandwidth of 4.2 MHz is required for video transmission.
32. A bandwidth of 6 MHz is required for TV signal because it transmits both voice and video.
33. Digital signals are in the form of rectangular waves.
34. Rectangular waves can be decomposed into a superposition of sinusodial waves of frequencies.