which set of integers is closed under multiplication

Question

negative integers

integers<5

positive integers

integers >-10

 

Answer 1

Closed operations means, that when you multiply ANY two elements

of the set, the result is also a member of the set.

 

Negative integers.

-------------------

NO! It is NOT closed.

The product of two negative integers is positive.

Ex. -2 x -1 = +2 <--- not negative. Not closed.

 

integers less than 5

----------------------

If we multiply ANY two integers less than

5, do we still get an integer less than 5?

NO!

 Here's a counter-example:

   -10 x -2 = +20

 

Multiplication is not closed on the set of integers less than 5.

Surely you can think of more counterexamples of your own.

 

Positive Integers

------------------

Yes, multiplication is a closed operation

on the set of positive integers. The product

of two positive integers MUST be a positive integer

 

Integers greater than -10

-------------------------------

 -9 > -10 and 2 > -10.

 But (-9)(2) = -18 < -10.

 

 -5 > -10 and 100> -10.

 But (-5)(100) = -500 < -10.

 

In fact, -1>-10 and multiplying by any number

greater than 10 by -1 will result in a product

that is less than or equal to -10.

 

Not even close!

 

Not closed.

Create your own counter examples.

 

 

Answer 2

 A positive integer times a positive integer is a a positive integer.

Positive integers are closed under multiplication.