SOLUTION: let R* be the set of all real numbers except 0. define * on R* by letting a*b = |a|b. a) show that * gives an associative binary operation on R*. b)show that there is a left iden

SOLUTION: let R* be the set of all real numbers except 0. define * on R* by letting ab = |a|b. a) show that gives an associative binary operation on R*. b)show that there is a left iden

Question 44378This question is from textbook
: let R* be the set of all real numbers except 0. define * on R* by letting a*b = |a|b.
a) show that * gives an associative binary operation on R*.
b)show that there is a left identity for * and a right inverse for each element in R*.
c)is R* with this binary operation a group?
d) explain the significance of this exercise. This question is from textbook

Answer by venugopalramana(3286) (Show Source): You can put this solution on YOUR website!
let R* be the set of all real numbers except 0. define * on R* by letting a*b = |a|b.
a) show that * gives an associative binary operation on R*.
TST (A*B)*C=A*(B*C)
LHS=(|A|B)*C=||A|B|C......WHETHER A AND B ARE +VE OR -VE ,THE FINAL ANSWR IS
|AB|C
RHS=A*(|B|C)=|A||B|C...HERE TOO THE FINAL ANSWER IS |AB|C
HENCE ASSOCIATIVE PROPERTY IS PROVED.
b)show that there is a left identity for * and a right inverse for each element in R*.
TST THERE IS X WHERE ....X*A=A
X*A=|X|A=A...THAT IS X=1 IS THE LEFT IDENTITY
BUT A*1=|A|1 IS NOT EQUAL TO A IF A IS -VE.HENCE X=1 IS ONLY LEFT IDENTITY AND NOT A RIGHT IDENTITY.
RIGHT INVERSE...LET A*A'=1
A*A'=|A|A'=1..SO A'=1/|A|WHICH EXISTS SINCE A IS NOT EQUAL TO ZERO AND IS INDEPENDENT OF SIGN OF A
LEFT INVERSE....A'*A=1=|A'|A=1....|A'|=1/A..WHICH EXISTS IF A IS POSITIVE.,BUT DOES NOT EXIST IF A IS NEGATIVE SINCE WE CANT FIND A' SUCH THAT |A'| IS NEGATIVE.
c)is R* with this binary operation a group?
WE HAVE TO PROVE
1.CLOSOURE....A*B IS AN ELEMENT IN R*...WHICH IS OK.SINCE A/B CANNOT BE ZEROS AS GIVEN
2.ASSOCIATIVE...PROVED ALREADY
3.EXISTENCE OF IDENTITY...WE HAVE 1*A=A...BUT A*1 IS NOT EQUAL TO A .HENCE THIS IS NOT A GROUP.
d) explain the significance of this exercise.
IT SHOWS THAT THERE COULD BE AN IDENTITY ELEMENT WHICH IS PARTIAL IN THE SENSE THAT IT IS ONLY LEFT IDENTITY AND NOT RIGHT IDENTITY.
RELATED QUESTIONS
let s be the set of all real numbers except -1. define * on s by a*b=a+b+ab. a)show that (answered by venugopalramana)

1. Show that the set of real numbers is dense 2. Show that the set of odd positive... (answered by EdenWolf)

the operation * defined on the set of real numbers by a*b=a+b+(ab)/2 for all a,b�R. is... (answered by rothauserc)

define a binary * on R*=R without 0 by a*b =ab if > 0 and a/b if a<0. Determine whether (answered by richard1234)

Let # be the binary operation defined on the positive real numbers by a#b=a^b. Find the... (answered by josgarithmetic)

1) Let R and S be rings and define R x S = {(r,s):rES} to be the set of all pairs (r,s)... (answered by kev82)

1)Given the binary operation * defined over the set of real numbers R by a*b=a+b+ab. What (answered by venugopalramana)

Show that the mapping f:R-R be defined by f(x) = ax + b where a,b � R, a is not equal to... (answered by rothauserc)

Can you help me solving this Q , I have problem with understand it Let A be the set... (answered by mathmate)