Question

What is a closure property? with respect to addition?

Answer 1

I understand commutative and associative but i'm unable to understand the closure, even after looking at the examples.

Answer 2

It's dealing with a slightly arcane but logically important concern.

What if when you add up two numbers you don't get a real number?

The closure property of addition says: "When you add up two real numbers, the result is also a real number."

So yes Virigina, 2 + 3 is also a real number, we call it five! And
\[ \sqrt{2-\sqrt{2}} + 1 \]
is also a real number!

As I say, it's logically important, and the idea of closure under operations like addition is really important later on in mathematics when we're talking about algebraic constructions that aren't the real numbers. But here, it just means that you don't have to worry about somehow working yourself into a bizarre situation when add up two numbers.

Answer 3

Hi James, but I have been given these options to choose from:
1. x+y=y+x for all real numbers x, y
2. x+(y+z)=(x+y)+z for all real numbers xyz
3. x-y=y-z for all reall numbers xy
4. none of these

to me the option 1 seems commutative, theoption 2 seems associative.

Answer 4

#3 is false.

The answer is 4.

Answer 5

A statement of closure under addition would be:
"For all real numbers x and y, x + y is also a real number"

Answer 6

Thank you so much James for the help.