SOLUTION: let s be the set of all real numbers except -1. define * on s by a*b=a+b+ab. a)show that * gives a binary operation on s. B)show that (s,*) is a group. c) find the solution of t

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Question 44377This question is from textbook
: let s be the set of all real numbers except -1. define * on s by a*b=a+b+ab.
a)show that * gives a binary operation on s.
B)show that (s,*) is a group.
c) find the solution of the equation 2*x*3=7 in s. This question is from textbook

Answer by venugopalramana(3286) About Me  (Show Source):
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let s be the set of all real numbers except -1. define * on s by a*b=a+b+ab.
a)show that * gives a binary operation on s.
CLOSOURE..A*B=A+B+AB..IS OBVIOUSLY AN ELEMENT OF S IF A AND B ARE ELEMENTS OF S
A*B CAN NOT BE -1...IF A AND B ARE NOT EQUAL TO -1.BECAUSE...IF
A*B=A+B+AB=-1...THEN...B(1+A)=-1-A=-(1+A)
B=-(1+A)/(1+A)=-1....AND....A(1+B)=-1-B=-(1+B)
A=-(1+B)/(1+B)=-1...BOTH OF WHICH ARE NOT POSSIBLE AS GIVEN
B)show that (s,*) is a group.
1.CLOSOURE ..PROVED ABOVE
2.ASSOCIATIVE...TST..(A*B)*C=A*(B*C)
LHS=(A+B+AB)*C=A+B+AB+C+(A+B+AB)C=A+B+C+AB+BC+AC+ABC
RHS=A*(B+C+BC)=A+B+C+BC+A(B+C+BC)=A+B+C+BC+AB+AC+ABC=LHS
ASSOCIATIVE IS PROVED
3.IDENTITY
A*X=A.....A+X+AX=A....X(1+A)=0...SINCE A CANNOT BE -1...X=0
X*A=X+A+XA=A..IS TRUE FOR X=0..SO X=0 IS THE IDENTITY ELEMENT
4.INVERSE
A*A'=0=A+A'+AA'
A'=-A/(1+A)..EXISTS AS A IS NOT EQUAL TO -1
SIMILARLY WE CAN SHOW THAT A'*A=0 ALSO GIVES US SAME A'=-A/(1+A)
HENCE THIS IS A GROUP.
c) find the solution of the equation 2*x*3=7 in s.
(2+X+2X)*3=2+X+2X+3+(2+X+2X)3=7
2+X+2X+3+6+3X+6X=7
12X=7-11=-4
X=-4/12=-1/3