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Operations on integers

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▪ I is closed for addition.

i.e. a $\in$ I, b $\in$ I $\Rightarrow$ a + b $\in$ I.

▪ (a + b) + c = a + (b + c) : for all a, b, c $\in$ I. [Associative Law]

▪ a + b = b + a : for all a, b $\in$ I. [Commutative Law]

▪ 0 is the additive identity in I, since

0 + a = a + 0 = a for all a $\in$ I.

▪ For each a $\in$ I there exists – a $\in$ I such that

a + (– a) = (– a) + a = 0.

Integer ‘– a’ is called the additive inverse of ‘a’.

▪ I is closed for subtraction, since

a $\in$ I, b $\in$ I $\Rightarrow$ a – b $\in$ I

▪ In general, a – b $\ne$ b – a, as 2 – 3 $\ne$ 3 – 2.

▪ In general, (a – b) – c $\ne$ a – (b – c) as (2 – 3) – 4 $\ne$ 2 – (3 – 4).

▪ I is closed for multiplication but not closed under division.

▪ Multiplication on I is commutative as well as associative. But not for division.

▪ Multiplication distributes addition on I. But not for division.

▪ 1 is the multiplicative identity.

Let a, b I. We say that ‘a’ is a factor of ‘b’, if there exists an integer ‘p’ such that b = ap. In this case, we write, a | b.

If ‘a’ is a factor of ‘b’, then ‘b’ is called a multiple of ‘a’.

: for any two integers a, b we say that a > b, if a = b + c for some integer c.

If a > b, then we write, b < a.

For any two integers ‘a’ and ‘b’ one and only one of the following holds :

(i) a > b or (ii) a < b or (iii) a = b