This is the standard construction of the integers from the natural numbers, with addition and multiplication.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
(i) Is (2,6) related to (4,8)? Give three ordered pairs which are related to (2,6)
(ii) Is (2,6) related to (a, a+4)? Justify your answer
(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)
OK, this is actually easier than you think.Originally Posted by alexis
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.(i) Is (2,6) related to (4,8)? Give three ordered pairs which are related to (2,6)
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.
Check this the same way as (i).(ii) Is (2,6) related to (a, a+4)? Justify your answer
You must prove three things:(iii) Prove (i.e. using general letters) that p is an equivalence relation on the set AxB
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
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)