PLEASE GO THROUGH THE FOLLOWING WAY FOR DOING QUESIONS ON GROUPS
For proving the questions on Groups many students commits mistakes on the following questions.
1. If Q1 is the set of rationall numbers other than 1 with binary operation * defined by a*b=a+b-ab for all a,b εQ1, Show that (Q1,*) is an abelian group and solve 5*x=3 in Q1.
2.If Q-1 is the set of all rational numbers except -1 and * is a binary operation defined on Q-1 by a*b= a+b+ab, Prove that Q-1 is an abelian group.
For verifying closure law for the above the problems, many students simply do in the following way
Closure law: a*b=a+b-ab is again a rational number and hence closure law is true. However Here student did not verified a*b is a rational number except 1. Hence it is very necessary to show that a*b is a rational number other than 1 . Other wise 1Mark will be reduced in the valuation. In april 2002, the 1 Mark is reduced for the students who did not verified the closure law correctly.
The correct approach is as follows:
For question number 1:
Let a, b Q1 such that a#1, b#1 a+b-ab is also a rational number and it cannot be equal to 1
Because If a+b-ab=1, then a+b-ab-1=0 (a-1)(b-1)=0 a=1 or b=1 which is not true as a#1 and b#1. a*bQ1 Q1 is closed w.r.t * (GIVING REASONS IS IMP)
Similarly for verfying closure for 2 question
Closure law: or Q-1 a*b=a+b-ab is againa a rational number and hence closure law is true.
Let a, b Q-1such that a# -1 b#-1 a+b+ab is also a rational number and it cannot be
equal to -1
Because If a+b+ab=-1, then a+b+ab+1=0 (a+1)(b+1)=0 a= -1 or b= -1 which is not true as a# -1 and b# -1. a*bQ-1 Q1 is closed w.r.t * (GIVING REASONS IS IMP)