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
2026-03-29 14:17:16.1774793836
Binary Operations for grouping
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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.