Some Doubt

In Mathematics Textbook for Class XII of N.C.E.R.T 2006 edition, question no. 8 Exercise 1.1 reads as:

Show that the relation R in the set A = {1,2,3,4,5} given by
R = {(a,b) : Ia-bIis even} is an equivalence relation. Show that all the elements of {1,3,5} are related to each other and all the elements of {2,4} are related to each other. But no element of {1,3,5} is related to any element of {2,4}.

My doubt is:

Can we say that "the relation R in the set A = {1,2,3,4,5} given by
R = {(a,b) : Ia-bIis even} is an equivalence relation"?

In fact this will be true if relation R is reflexive. That is if a R a for all a belonging to A.

Which means that 1 - 1 = 0 is even.

Can we say ' 0 is even'?

Please comment.

Play with Numbers

Let us play with numbers:

Guide lines to play the game: Example

Ask your friends to

1. Select any three non zero digits. Say 5, 3, 7

(without telling you)

2. Form all possible two-digit numbers from 53

the three digits selected by them. 57

(they will always get 6 such numbers) 35

37

75

73

3. Add the six numbers obtained in step 2. 330

4. Add the three digits selected in step 1. 5+3+7=15

5. Divide the sum obtained in step 3 by the 15 330 15 = 22

sum obtained in step 4.

6. You may tell them the answer quickly since it

will always be 22.

You may repeat this game by asking your friends to form there digit numbers in step 2 (in stead of 2 digit numbers). In this case answer will always be 222.

Number pattern

Observethefollowing pattern and write next two rows.

6x7 = 42

66x 67 = 4422

666x 667 = 444222

6666x6667 = 44442222

-------------- = --------------

--------------- = ------------------

Verify the pattern by multiplying actually

For a change

For a change

Look at the following special number: 381654729 Try to guess “Why it has been called a special number?” It is because of its special characteristics listed below: It is a 9-digit number comprising of the 9-digits of our number system, with no repetition. Number formed by its one digit from the left is divisible by 1. Number formed by its two digits from the left is divisible by 2. 38divded by 2 is equal to 19 Number formed by its three digits from the left is divisible by 3. 381divded by 3 is equal to 127 Number formed by its first four digits from the left is divisible by 4. 3816 divided by 4 is equal to 954 And so on up to: Number formed by its all the nine digits from the left is divisible by 9. 381654729 divided by 9 is equal to 42406081 Is it not a special number? Can you form some other number with the characteristics listed above?

Let us play with Mathematics