# sets

• Oct 22nd 2005, 04:30 AM
alexis
Right then who can do this tuff cookie (2)
2. Let AxB be the set of ordered pairs (a,b) where a and b belong to the set of natural numbers N.

A relation p: AxB ----> AxB is defined by: (a,b)p(c,d) <-----> a+d = b+c

(i) Is (2,6) related to (4,8)? Give three ordered pairs which are related to (2,6)

(iii) Prove (i.e. using general letters) that p is an equivalence relation on the set AxB

EXTENSION

(iv) As p is an equivalence relation there are accociated equivalece classes. Find all the ordered pairs in the equivalence class of (2,6). Why could this equivalence class be identified with the integer -4?

(v) Give the equivalence classes (as sets of ordered pairs) defined by p for each of the integers: 0, -1 and +1

(vi) Consider two general ordered pairs, (a,b) and (c,d). If addition is defined by (a,b) + (c,d) = (a+c, b+d) and multiplication is defined by (a,b) x (c,d) = (ac+bd, ad+bc), show that these definitions provide a way of demonstrating that (+1) + (-1) = 0 and (-1) x (-1) = (+1)
• Oct 23rd 2005, 05:55 AM
hpe
Quote:

Originally Posted by alexis
2. Let AxB be the set of ordered pairs (a,b) where a and b belong to the set of natural numbers N.

A relation p: AxB ----> AxB is defined by: (a,b)p(c,d) <-----> a+d = b+c

This is the standard construction of the integers from the natural numbers, with addition and multiplication.
• Oct 23rd 2005, 02:49 PM
niva
hi hpe

can you explain how you construct? I mean I wanna know the details of the question vi.
• Oct 23rd 2005, 02:58 PM
alexis
And I still would love to understand each question, I am pretty bad at Sets :( as I can't even start the coursework!

I mean a little hint for each question would be just fantastic and I think I might be able to do the rest :confused:
• Oct 23rd 2005, 06:18 PM
hpe
Quote:

Originally Posted by alexis
And I still would love to understand each question, I am pretty bad at Sets :( as I can't even start the coursework!

I mean a little hint for each question would be just fantastic and I think I might be able to do the rest :confused:

OK, this is actually easier than you think.
Quote:

(i) Is (2,6) related to (4,8)? Give three ordered pairs which are related to (2,6)
So here a = 2, b = 6, c = 4, d = 8. Then a+d = 2+8 = 10 and b + c = 4 + 6 = 10, so these two number pairs are indeed related.

To find others, write the condition as follows: a - b = c - d (so that the values of one pair appear on the left and the values of the other on the right). Here a - b = 2 - 6 = -4. Therefore any pair (c,d) with c-d = -4 (or d-c = 4) is also related. Pick three pairs (c,d) with this property.
Quote:

Check this the same way as (i).
Quote:

(iii) Prove (i.e. using general letters) that p is an equivalence relation on the set AxB
You must prove three things:

a) Any number pair is related to itself, (a,b) ~ (a,b). So you must prove that the first number on the left plus the second on the right, a+b, is the same as the corresponding sum, first n the right plus second on the left. You should find that this is obvious.

b) If (a,b), (c,d) are number pairs and (a,b) ~ (c,d), then you must show that (c,d) ~ (a,b). Now write down what the assumption means. It means a+d = b + c. Then write down what the conclusion means. It means c + b = d + a. Does this follow from the assumption? Take a look.

c) The third property is transitivity. Start with three number pairs (a,b), (c,d), (e,f) and assume (a,b)~(c,d) and (c,d) ~ (e,f). You must prove (a,b) ~ (e,f). Now write down what the first part of the assumption means: It means a + d = b + c (as in part b)). Write down the corresponding equation for the second part of the assumption. Finally write down the equation that corresponds to what you must prove, with one loine of space in between. Take a look - does the conclusion follow from the two assumptions?

Hope this helps :)
• Oct 25th 2005, 07:59 AM
alexis
Thanks Hpe!

For c) is the answer then:

From (a,b) p (c,d) we get a+d = b+c
From (c,d) p (e,f) we get c+f = d+e

As (c,d) p (c,d) we know that c+d = c+d

Therefore a+d = d+e
Hence (a,b) p (e,f) :)

Any hints for these questions?

(iv) As p is an equivalence relation there are associated equivalence classes. Find all the ordered pairs in the equivalence class of (2,6). Why could this equivalence class be identified with the integer -4?

(v) Give the equivalence classes (as sets of ordered pairs) defined by p for each of the integers: 0, -1 and +1

(vi) Consider two general ordered pairs, (a,b) and (c,d). If addition is defined by (a,b) + (c,d) = (a+c, b+d) and multiplication is defined by (a,b) x (c,d) = (ac+bd, ad+bc), show that these definitions provide a way of demonstrating that (+1) + (-1) = 0 and (-1) x (-1) = (+1)