Existence of identity; existence of inverse and cancellation laws

Define an operation * on Q as follows: a * b =((a+b)/(2)), a, b in Q . Examine the existence of identify and existence of inverse for the operation * on Q.

(i) Let M={({:(x,x),(x,x):}):x in R -{0}} and let ** be the matrix multiplication. Determine whether M is closed under ** . If so, examinie the existence of identity, existence of inverse properties for the operation ** on M.

Verify (i) closure property (ii) commutative property (iii) associative property (iv) existence of identity and (v) existence of inverse for the operation +_(5) on ZZ_5 using table corresponding to addition modulo 5.

Verify (i) closure property (ii) commutative property (iii) associative property (iv) existence of identity and (v) existence of inverse for the operation xx_(11) on a subset A = {1,3,4,5,9} of the set of remainders {0,1,2,3,4,5,6,7,8,9,10}.

Existence of additive identity

Existence of additive inverse

Existence of negative (additive inverse) of a rational number