Binary Operations for grouping Which of the following binary operations are closed?


*

*subtraction of positive integers

*division of nonzero integers

*function composition of polynomials with real coefficients 

*multiplication of $2\times 2$ matrices with integer entries 

 A: Let's start with the first one.  Suppose that $a,b$ are each positive integers.  Are we sure that $a-b$ is always a positive integer?  In fact, no.  For example, if $a=2$ and $b=10$, both positive integers, then $a-b$ is no longer a positive integer.  Hence the set of positive integers is not closed under subtraction.
A: To continue the answer that vadim123 provided:
Division of nonzero integers is not closed. Here is how we know. Suppose that a,b are each nonzero integers. Is it guaranteed that a/b is an integer? Well, to prove that the division of nonzero integers is not closed, you only need to enumerate one example that demonstrates an integer a/b not being in the set of integers. One such example would be when a = 5 and b = 7. The value  5/7 is not inside the set of nonzero integers. Therefore, division of nonzero integers cannot possibly be closed. 
Multiplication of 2x2 matrices with integer entries is closed. (Informally) We know that it is closed because the multiplication of square matrices always produces another square matrix of the same size and because the integer entries inside those matrices are closed under addition and multiplication. 
