Prove the following is a group

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In summary, to prove that a nonempty set S with an associative binary operation * on S, where a*c=b and d*a=b have solutions in S for all a,b in S, is a group, the following must be shown: associativity of *, existence of identity element and inverse for each element in S. By using the given premise, it is possible to show the existence of an identity element and inverses for all elements in S. This can be achieved by picking the same element a for both a and b, and using the given solutions c and d to prove the existence of an identity element, and then using this identity element to prove the existence of inverses for all elements in S.
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Homework Statement


Prove that a nonempty set S with an associative binary operation * on S such that a*c=b and d*a=b have solutions in S for all a,b in S is a group.


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The Attempt at a Solution


In order to prove something is a group, I know that I must show associativity of *, existence of identity element and inverse for each element in S.
- We are given that S has an associative binary operation, so that is satisfied.
- In regards to finding the identity element such that a*e=e*a=a for all a in S...this is where I am not sure where to begin. Any suggestions on how to start?
 
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Take a and b to be the same element a. Then your premise tells you you can find c and d such that a*c=a and d*a=a. It looks like c and d should be your identity 'e', right? To show that's true, you have to prove c=d, and you have to prove if you pick another element besides a, you get the same identity. It can be done. You just have to keep using the only premise you've got over and over. Similar course for inverses.
 

1. What are the four requirements for a set to be considered a group?

In order for a set to be considered a group, it must have closure, associativity, identity element, and inverse element. Closure means that the operation performed on any two elements in the set will result in an element that is also in the set. Associativity means that the order in which the operations are performed does not matter. The identity element is an element in the set that, when combined with any other element, will result in that same element. The inverse element is an element in the set that, when combined with another element, will result in the identity element.

2. How do you prove that a set satisfies the closure property?

In order to prove that a set satisfies the closure property, you must show that for any two elements in the set, the operation performed on them will result in an element that is also in the set. This can be done by using a specific example or by showing that the operation is closed under all possible combinations of elements in the set.

3. Can a set have more than one identity element?

No, a set can only have one identity element. This is because the identity element must satisfy the property that when combined with any other element, it will result in that same element. If there were more than one identity element, this property would not hold true.

4. How do you prove associativity for a set?

To prove associativity for a set, you must show that for any three elements in the set, the order in which the operations are performed does not matter. This can be done by using a specific example or by showing that the operation is associative for all possible combinations of elements in the set.

5. What is the difference between a group and a groupoid?

A group is a set that satisfies the four requirements of closure, associativity, identity element, and inverse element. A groupoid is a set that satisfies the closure and associativity properties, but may not have an identity element or inverse element. In other words, a groupoid is a less strict version of a group.

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