. . Some more properties of a binary operation Commutative property Any binary operation ∗ defined on a nonempty set S is said to satisfy the commutative property, if a b b a a b S ∗ ∗ ∀ , Number System Operation - - Associative property Any binary operation ∗ defined on a nonempty set S is said to satisfy the associative property, if b c a b a b c S ∗ ∗ ∗ ∗ ∀ , , Existence of identity property An element e S is said to be the Identity Element of S under the binary operation ∗ if for all S we have that a e ∗= and e a ∗ Existence of inverse property If an identity element e exists and if for every a S , there exists b in S such that a b ∗ and b a ∗ then b S is said to be the Inverse Element of a . In such instances, we write b − .
Note a – is an element of S . It should be read as the inverse of a and not as a . Note (i) The multiplicative identity is in and it is the one and only one element with the property ⋅= ⋅ ∀∈ . (ii) The multiplicative inverse of any element, say in is and no other nonzero rational number x has the property that ⋅ ⋅ Note Whenever a mathematical statement involves ‘for every’ or ‘ for all’ , it has to be proved for every pair or three elements.
It is not easy to prove for every pair or three elements. But these types of definitions may be used to prove the negation of the statement. That is, negation of “for every” or “for all” is “there exists not”. So, produce one such pair or three elements to establish the negation of the statement.
The questions of existence and uniqueness of identity and inverse are to be examined. The following