Could you help me Determine whether each set of 
numbers is closed under the indicated operation
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Irrational,addition

The question is 

"Can you add your way out of the set of
irrational numbers?  

The answer is "Yes you can."  

 is irrational and so is 
but if you add them you get , which is
rational, so you have added your way out of 
the irrational numbers. That's means it's NOT 
CLOSED.   
-----------------------
natural,addition

The question is 

"Can you add your way out of the set of
natural numbers?  

The answer is "No you can't".  If you add any two 
natural numbers you always get another natural number.  
That means it is CLOSED.
-----------------------
irrational,multiplcation

The question is 

"Can you multiply your way out of the set of
irrational numbers?  

The answer is "Yes you can."  

 is irrational and so is 
but if you multiply them you get , which is 
rational, so you have muliplied your way out of the 
irrational numbers. That's means it's NOT CLOSED.  
-----------------------
integers,subtraction

The question is 

"Can you subtract your way out of the set of
integers?  

The answer is "No you can't".  If you subtract any two 
integers you always get another integer.  
That means it is CLOSED.

-----------------------
rational, multiplcation

The question is 

"Can you multiply your way out of the set of
rational numbers?  

The answer is "No you can't".  If you multiply any two 
rational numbers you always get another rational number.  
That means it is CLOSED.

Edwin