MathCS.org - Real Analysis: Examples 1.4.3(c):

Examples 1.4.3(c):

Let A be the set N x N and define an equivalence relation r on N x N and addition of the equivalence classes as follows:

(a,b) is related to (a’,b’) if a + b’ = a’ + b
[(a,b)] + [(a',b')] = [(a + a', b + b')]
[(a,b)] * [(a’, b’)] = [(a * b’ + b * a’, a * a’ + b * b’)]

What is the best symbol to use for the resulting equivalence classes ?

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Since two pairs (a,b) and (a', b') are related if

a + b' = a' + b or b - a = b' - a'

we might as well choose the symbol b - a to denote their equivalence classes. Hence:

the symbol 2 denote the equivalence class [(1,3)] containing, for example, the pairs (1,3), (5,7), and (100, 102).
the symbol -3 denotes the equivalence class [(4,1)], containing, for example, the pairs (4,1), (8,5), and (103, 100).

By the above rules, if the symbols 2 and -3 are added together we get the class

2 + -3 = [(1,3)] + [(103,100)] = [(1,2)] = -1

and if the symbols 2 and -3 are multiplied together we get the class

2 * (-3) = [(1,3)] * [(103,100)] = [(1,7)] = -6

Hence, these equivalence classes, together with the definition of addition and multiplication, give a mathematically precise meaning to the symbol -2, and explains in fact the meaning, the addition, and the multiplication of the integers Z

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Examples 1.4.3(c):