Group theory

Question

Which of the following operation on A is closed;

 where set A={0,1,2,3,4,5,6,7}?

 

a) a*b=abmod10       

b)a*b=|a-b|mod10

c) a*b=(a^2 + 2ab + b^2) mod 10         

 d) a*b=(7a+b) mod 10

Comments

Answer: B

A set has closure under an operation if performance of that operation on members of the set always produces a member of the same set; in this case we also say that the set is closed under the operation.

a) When a=4 and b=2, a*b=8 mod 10=8 which does not belong to A hence not closed

b) a*b is closed since even the farthest values of a  and b returns value 7 which belongs to the set A.

c) When a=3 b=4, a*b=49 mod 10 =9 which does not belong to the set A. Hence not closed.
d) When a=1 and b=2, a*b=9 mod 10=9 which does not belong to set A, hence not closed.