Binary operations, subsets and closure

In summary, "Closed under an operation" means that if a and b are in the set then a*b is also in the set, and has nothing to do with the topological notion of "closed set". The set of integers is closed under addition, whereas the set of even integers is also closed under addition, but the set of odd integers is not closed under addition.
  • #1
icantadd
114
0

Homework Statement



1)
Let S be a set and p: SxS->S be a binary operation. If T is a subset of S, then T is closed under p if p: TxT->T. As an example let S = integers and T be even Integers, and p be ordinary addition. Under which operations +,-,*,/ is the set Q closed? Under which operations +,-,*,/ is the set of irrationals closed

Homework Equations



not sure

The Attempt at a Solution


for 1) the irrationals can't be closed under any operations. Q is probably closed under all of them except division. But it is the example and definition that gets me. If T is a subset of the Integers, there is no way it will be closed under addition. Just take the Sup of the set and add 1 or the inf of the set and subtract 1. So I guess my question is: is closure peculiar to the subset we have taken, and if so How can we say that an entire set, like the Integers is closed under some operation.
 
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  • #2


The even integers are a subset of the integers and they are closed under addition, aren't they?
 
  • #3


icantadd said:

Homework Statement



1)
Let S be a set and p: SxS->S be a binary operation. If T is a subset of S, then T is closed under p if p: TxT->T. As an example let S = integers and T be even Integers, and p be ordinary addition. Under which operations +,-,*,/ is the set Q closed? Under which operations +,-,*,/ is the set of irrationals closed

Homework Equations



not sure

The Attempt at a Solution


for 1) the irrationals can't be closed under any operations. Q is probably closed under all of them except division. But it is the example and definition that gets me. If T is a subset of the Integers, there is no way it will be closed under addition. Just take the Sup of the set and add 1 or the inf of the set and subtract 1. So I guess my question is: is closure peculiar to the subset we have taken, and if so How can we say that an entire set, like the Integers is closed under some operation.
"Closed under an operation", *, means that if a and b are in the set then a*b is also in the set. That has nothing to do with sup or inf. You may be confusing this, algebraic, notion of "closed" with the topological notion of "closed set". They have nothing to do with one another.

If you add two integers the result is again an integer so the set of integers is closed under addition. If you add two even integers together, the result is again an even integer so the set of even integers, as Dick said, is closed under addition. The sum of two odd integers is even so the set of odd integers is not closed under addition.
 

1. What are binary operations?

Binary operations are mathematical operations that involve two operands, or inputs, to produce a single result. Common examples include addition, subtraction, multiplication, and division.

2. How do binary operations relate to subsets?

In the context of sets, binary operations are often used to combine elements from two subsets to create a new subset. For example, the union of two sets is a binary operation that combines all elements from both sets into a new set.

3. What is the closure property in binary operations?

The closure property states that performing a binary operation on two elements from a set will always result in another element within that same set. In other words, the result of a binary operation on elements from a set will always remain within that set.

4. Can binary operations be non-commutative?

Yes, binary operations can be either commutative or non-commutative. Commutative operations produce the same result regardless of the order of the operands, while non-commutative operations do not.

5. How are binary operations used in computer science?

Binary operations are fundamental to computer science and are used in various applications, such as data compression, cryptography, and digital logic. In computer programming, binary operations are commonly used to manipulate binary numbers and perform logical operations on binary data.

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