Can someone explain to me what closure property is and how to identify it from an equations?

- anonymous

Can someone explain to me what closure property is and how to identify it from an equations?

- schrodinger

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- Directrix

Do you have a specific problem in mind? If so, please post it.

- anonymous

Not really. I have a math teacher that assumed we learned this already and it's not in the math textbook. I just need help with the problems where you identify the property ex. 3+(4+5)=3+(5+4) Commutative property

- Directrix

Start with the properties attached in the file.

##### 1 Attachment

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- Directrix

After you read them, go to the practice site at the following link:
http://www.regentsprep.org/Regents/math/ALGEBRA/AN1/propPrac.htm

- anonymous

I know these properties already. The only property I don't understand is the closure property

- Directrix

Okay, I want to ask you to answer this question first. Thanks.
Name the property: For real numbers, a, b, and c, give the property that justifies the following:
a + (b + c) = a + (c + b)

- anonymous

Commutative Property because the values contained in the parenthesis stay the same, right?

- Directrix

Yes, very good. Most people focus on the three elements and answer "associative."
Now for closure.
See the two attached files.

##### 2 Attachments

- anonymous

Done. How would the problems look like?

- Directrix

Question 1: Is the set of negative integers closed under the operation of multiplication?
Questions 2: Is the finite set { -1, 0, 1 } closed under the operation of division?

- anonymous

Q1: No
Q2: No

- anonymous

But are there problems that look like For all real numbers a and b, {x|x.....?
My teacher seems to love those.

- Directrix

Your answers to Q1 and Q2 are correct.
That is set notation your teacher is using. If you would look in your notes and find one of those problems and post it here, that would be good. We could discuss it.

- anonymous

I'm not sure if I copied it down, but I'll check >.<

- anonymous

I can't find it, but the problems after the closure one all began with something like|dw:1347588986815:dw|

- Directrix

It's always good to take notes even if you understand what is being said at the time. I'll look in my books for a problem in that form. Question: Do you know the components of set builder notation?

- anonymous

Yep.

- Directrix

So the two examples in the attached file are familiar? Yes?

##### 1 Attachment

- Directrix

What does the symbol on the attached file mean?

##### 1 Attachment

- anonymous

Or does not

- Directrix

Right. So, the question I have is "What specifically is your question regarding set notation and closure.?" It seems that you know the lingo.

- anonymous

I have a test tomorrow and I'm just afraid that the closure problem will be something completely unexpected so I wanted to find someone who might know.

- Directrix

If you had an example of just one problem which you have in mind as a "closure problem," then that would be a good place to start.

- anonymous

In which lies the problem...
One more thing. Is: \[\sqrt{2}+3 is a real number\]
an example of the Closure Property of Addition?

- anonymous

@Directrix . Last thing. I promise.

- Directrix

The math processor is at 0% so I can't read the equation editor items with clarity. I think the question is the following: "(Is square root of 2) + 3 a real number. Yes, it is because square root of 2 is a real number and 3 is a real number and the set of real numbers is closed under the operation of addition.
Look over your notes and ask other questions if you want. I don't mind.
Question for you: Is the set of imaginary numbers closed under the operation of subtraction?

- anonymous

I don't think so.

- anonymous

I just know that i-i=0, but I'm not sure if you can repeat the elements of the set

- Directrix

They are not. A counterexample to the statement that the set of imaginary numbers is closed with respect to subraction would be the following: 5i and 5i are imaginary numbers but 5i - 5i = 0i = 0 which is not an imaginary number.
Question: Is the set of rational numbers closed with respect to multiplication?

- anonymous

Yes.

- Directrix

Correct.
Is the set of irrational numbers closed with respect to a) subtraction; b) multiplication?

- anonymous

a)no
b)no

- Directrix

Correct on both.
Question: The sum of the conjugate complex numbers (a+bi) and (a-bi) is a real number. Does this violate the closure property for addition of complex numbers?

- anonymous

No? I mean, they're both real and imaginary but the result is 2a, so I'd imagine that it wouldn't violate it o-o

- Directrix

The reals are a subset of the complex numbers so the closure property would not be violated. Your answer is correct.

- anonymous

Do I pass yet? :P

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