Right inverse, left inverse, binary operations

In summary, the conversation discusses the concept of left and right inverses in a binary operation on a set B, where the neutral element is denoted as p. It is stated that if an element in B has both a left and right inverse, they are equal. The conversation also touches on the importance of associativity in this concept, and the confusion caused by the use of arbitrary variables in the original definitions.
  • #1
Panphobia
435
13

Homework Statement


If * is a binary operation on a set B, and the domain of definition is B^2, if * is associative and the neutral element is p. If r and l are elements of b we can say that r is a left inverse of l under * iff r * l = p, and l is a right inverse of r iff l * r = p. Show that if an element of B has a left and right inverse, then they are equal.

The Attempt at a Solution


Does the neutral element have anything to do with finding the answer, also what does associativity have to do with finding the answer? All I can think of is since there is only one neutral element in *, r = p and l = p, but I don't think that is the answer.
 
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  • #2
Panphobia said:

Homework Statement


If * is a binary operation on a set B, and the domain of definition is B^2, if * is associative and the neutral element is p. If r and l are elements of b we can say that r is a left inverse of l under * iff r * l = p, and l is a right inverse of r iff l * = p. Show that if an element of B has a left and right inverse, then they are equal.


The Attempt at a Solution


Does the neutral element have anything to do with finding the answer, also what does associativity have to do with finding the answer? All I can think of is since there is only one neutral element in *, r = p and l = p, but I don't think that is the answer.

Sure, you have to assume there is a neutral element p, so that for any x, x*p=p*x=x. Now suppose r is a right inverse of x (so x*r=p) and l is a left inverse of x (so l*x=p). Associativity tells you (l*x)*r=l*(x*r), yes? So?
 
  • #3
Does it matter than the question say l * r = p and r * l = p, where does the x come from? would this mean (l * r) * l = l * (r * l)?
 
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  • #4
Panphobia said:
Does it matter than the question say l * r = p and r * l = p, where does the x come from? would this mean (l * r) * l = l * (r * l)?

The letters were confusing and I changed them. Your writing (l * r) * l = l * (r * l) shows why. The definitions you've got are using 'l' to be a symbol for both the right and left inverse. They are two different things. Those definitions mean something independent of the particular symbols used. Suppose I tell you that b is a right inverse of c. Let's keep the symbol 'p' to mean the neutral element, though I probably would have used the symbol '1' instead. It's more suggestive. What does that mean?
 
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  • #5
Then why in the question is l * r = p, and r * l = p defined? btw r and l do not mean right and left inverse, they are arbitrary variables, the question says, "l is left inverse of r under * iff l * r = p, and l is a right inverse of r under * iff r * l = p". It is kind of confusing.
 
  • #6
First off, look at some numbers you know, such as the rationals. With * being the multiplication operator you learned long ago and p being 1, what is the left inverse of 2 ? The right inverse? The answer is obviously 1/2 in both cases: 1/2*2 = 2*1/2 = 1. The goal of this problem is to show that if an element of B has both left and right inverses and if * is associative then there is only one inverse (i.e., the left and right inverses are one and the same).

Note that this result does not necessarily mean that * is commutative. What it does mean is that an element of B commutes with its multiplicative inverse.

Dick took the correct approach by giving that element of B that has both a left and right inverse a new symbol, x.
 
  • #7
Panphobia said:
Then why in the question is l * r = p, and r * l = p defined? btw r and l do not mean right and left inverse, they are arbitrary variables, the question says, "l is left inverse of r under * iff l * r = p, and l is a right inverse of r under * iff r * l = p". It is kind of confusing.

Yes, they are arbitrary variables, and yes, it's kind of confusing. The reason it's confusing is because they made such a poor choice of arbitrary symbols. Try to unscramble it. You didn't answer my question. I'll repeat it. Suppose I tell you that b is a right inverse of c. What does that mean? Bypass the whole 'l' and 'r' thing.
 
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  • #8
c * b = p, where p is the neutral element
 
  • #9
Panphobia said:
c * b = p, where p is the neutral element

Bingo! Now I'll tell you that d is a left inverse of c. Same question. And then can you show b=d?
 
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  • #10
(d * c) * b = d * (c * b)
d * c = c * b = p
p * b = d * p
so this implies d = b, right?
 
  • #11
Panphobia said:
(d * c) * b = d * (c * b)
d * c = c * b = p
p * b = d * p
so this implies d = b, right?

Right. Couldn't be clearer. I'm almost wondering if they didn't choose the symbol choice in original definitions to try to test you and throw you off track.
 
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  • #12
Thanks for the help!
 

Related to Right inverse, left inverse, binary operations

What is a right inverse?

A right inverse of an element in a binary operation is an element that, when combined with the original element, gives the identity element as the result. In other words, it "undoes" the operation on the right side.

What is a left inverse?

A left inverse of an element in a binary operation is an element that, when combined with the original element, gives the identity element as the result. In other words, it "undoes" the operation on the left side.

Can an element have both a left and a right inverse?

Yes, an element can have both a left and a right inverse in a binary operation. This is known as a two-sided inverse or a two-sided identity.

Do all binary operations have a left and a right inverse?

No, not all binary operations have a left and a right inverse. Only operations that are associative and have an identity element will have both a left and a right inverse for every element.

How do right and left inverses relate to each other?

In a binary operation, if an element has both a left and a right inverse, then they are equal and the operation is said to be invertible. If an element only has a left or a right inverse, then they are not necessarily equal and the operation is not invertible.

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