on . (ii) Also m m n ∀ . So the commutative property is satisfied (iii) ∀ m n p , , . Hence the associative property is satisfied.
(iv) m ⇒ = . Thus ∃ ⋺ ( . Hence the existence of identity is assured. (v) m ⇒ = − ∀ ∃− ' ' ' Thus ⋺ + − = − .
Hence, the existence of inverse property is also assured. Thus we see that the usual addition + on satisfies all the above five properties. Note that the additive identity is and the additive inverse of any integer m is - m . Example .
Verify the (i) closure property, (ii) commutative property, (iii) associative property (iv) existence of identity and (v) existence of inverse for the arithmetic operation - on . (i) Though - is not binary on ; it is binary on . To check the validity of any more properties satisfied by – on , it is better to check them for some particular simple values. (ii) Take m = , n = and ( = − and ( .
Hence ( ≠ . So the operation - is not commutative on . (iii) In order to check the associative property, let us put m and p = in both - - and m - - ) . −= −− =− …( ) .
…( ) From ( ) and ( ), it follows that m – – – – ≠ ) . Hence – is not associative on . (iv) Identity does not exist (why?). (v) Inverse does not exist (why?).
Example . Verify the (i) closure property, (ii) commutative property, (iii) associative property