Showing that a subset is closed under a binary operation

In summary, we need to show that H is closed under the associative binary operation *. To do so, we take two elements b and c from H and show that their product b*c is also in H. Using the given equations bx=xb and cx=xc, we can show that (bc)x = x(bc). This demonstrates that H is indeed closed under *.
  • #1
Mr Davis 97
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Homework Statement


Suppose that * is an associative binary operation on a set S. Let ##H=\{a \in S ~| ~a*x=x*a, ~ \forall x \in S\}##. Show that * is closed under H.

Homework Equations

The Attempt at a Solution


Let b and c be two different elements in H. We need to show that b*c is also in H.

We know that bx = xb, and that cx = xc. Putting these two equations, and using associativity and commutativity the farthest I can get is (bc)xx = xx(bc). I'm not sure how to get (bc)x = x(bc)
 
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  • #2
Mr Davis 97 said:

Homework Statement


Suppose that * is an associative binary operation on a set S. Let ##H=\{a \in S ~| ~a*x=x*a, ~ \forall x \in S\}##. Show that * is closed under H.

Homework Equations

The Attempt at a Solution


Let b and c be two different elements in H. We need to show that b*c is also in H.

We know that bx = xb, and that cx = xc. Putting these two equations, and using associativity and commutativity the farthest I can get is (bc)xx = xx(bc). I'm not sure how to get (bc)x = x(bc)
You start with ##(bc)x##. And it's " H is closed under * ".
 
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FAQ: Showing that a subset is closed under a binary operation

What does it mean for a subset to be closed under a binary operation?

When a subset is closed under a binary operation, it means that if any two elements in the subset are combined using the binary operation, the result will also be an element in the subset.

Why is it important to show that a subset is closed under a binary operation?

Showing that a subset is closed under a binary operation is important because it ensures that the binary operation can be applied to any two elements in the subset without resulting in an element outside of the subset. This is necessary for the subset to have the properties and behaviors that are expected from the binary operation.

How can I prove that a subset is closed under a binary operation?

To prove that a subset is closed under a binary operation, you can use the closure property, which states that the result of any operation between two elements in the subset will also be an element in the subset. This can be demonstrated by showing that any possible combination of elements in the subset using the binary operation will result in an element in the subset.

Can a subset be closed under one binary operation but not another?

Yes, a subset can be closed under one binary operation but not another. This is because different binary operations may have different closure properties, and what may be true for one operation may not be true for another. It is important to specify which binary operation you are referring to when discussing closure under a subset.

Are there any other properties that a subset must have in order to be closed under a binary operation?

Yes, there are other properties that a subset must have in order to be closed under a binary operation. In addition to the closure property, the subset must also contain the identity element for the binary operation and every element in the subset must have an inverse element. These properties ensure that the subset behaves as expected under the binary operation.

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