In Math, the whole numbers and negative numbers together are called integers. The set of all integers is denoted by Z.
Z = {... - 2, - 1,0,1,2, ...}, is the set of all integers
Here, we are going to see the following the three properties of subtraction of integers.
(i) Closure property
(ii) Commutative property
(iii) Associative property
Observe the following examples:
(i) 12 - 5 = 7
(ii) 5 - 12 = -7
(iii) (-18) - (-13) = -18 + 13 = -5
(iv) (-13) - (-18) = -13 + 18 = 5
(v) (-18) - 13 = -18 - 13 = -31
(vi) 18 - (-13) = 18 + 13 = 31
From the above examples, it is clear that subtraction of any two integers is again an integer.
In general, for any two integers a and b, a - b is an integer.
Therefore, the set of integers is closed under subtraction.
Consider the integers 7 and 4. We see that
7 - 4 = 3
4 - 7 = - 3
Therefore, 7 - 4 ≠ 4 - 7
In general, for any two integers a and b
a - b ≠ b - a
Therefore, we conclude that subtraction is not commutative for integers.
Consider the integers 7, 4 and 2
7 - (4 - 2) = 7 - 2 = 5
(7 - 4) - 2 = 3 - 2 = 1
Therefore, 7 - (4 - 2) ≠ (7 - 4) - 2
In general, for any three integers a , b and c
a - (b - c) ≠ (a - b) - c.
Therefore, subtraction of integers is not associative.
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