Question

Which of the following binary operations are closed? a. subtraction of positive integers b. division of nonzero integers c. function composition of polynomials with real coefficients d. multiplication of 2×2 matrices with integer entries ​

Answer 1

The goal is to determine whether or not the following binary operations are closed.

A. Subtraction of positive integers: Under normal substation operation, a set of positive integers is not near. Since 1∈Ζ⁺ and 2∈Ζ⁺ but (2-1) = ₋1∉Z⁺.

B. A division of nonzero integers: Non-zero integer set Z/{0} is not closed under the operation. Since 1∈Z/{0} and 2∈ Z/{0} but 1/2 ∉ Z/{0}.

C.  Function composition of polynomials with real coefficients: The set of polynomials with real coefficients is closed when using the typical binary operator. Use the obvious conclusion that a polynomial is a combination of two polynomials $f(x)$ and $g(x)$. Since every polynomial coefficient is real, every polynomial's composition will also have real coefficients.

D. The multiplication of 2×2 matrices with integer entries: Under the action of standard matrix multiplication Μ₂ₓ₂ (Z), the collection of matrices with integer elements, is close.

Learn more about Binary operation

https://brainly.ph/question/22621607

#SPJ4