which of the following binary operation are associative?

Question

1. Multiplication mod n.
2. Division of nonzero rationals.
3. Function composition of polynomials with real coefficients.
4. Multiplication of 2x2 matrices with integer entries.

Answer 1

1. $((x.y).z) = (x.(y.z)) =xyz \equiv r (mod\; n)$, associative
2. $\frac{\frac{1}{2}}{3}=\frac{1}{6} \neq \frac{3}{2} =\frac{1}{\frac{2}{3}}$ , not associative

3. $f(x)=\sum\limits_{i=0}^m{a_ix^i}$

   $g(x)=\sum\limits_{i=0}^n{b_ix^i}$

   $h(x)=\sum\limits_{i=0}^p{c_ix^i}$

   $(goh)(x)=\sum\limits_{i=0}^n{b_i\left(\sum\limits_{j=0}^p{c_jx^j}\right)^i}$

   $(fog)(x)=\sum\limits_{i=0}^m{a_i\left(\sum\limits_{j=0}^n{b_jx^j}\right)^i}$

   $(fo(goh))(x)=\sum\limits_{i=0}^m{a_i\left(\sum\limits_{j=0}^n{b_j\left(\sum\limits_{k=0}^p{c_kx^k}\right)^j}\right)^i}$

   $((fog)(x)=\sum\limits_{i=0}^m{a_i\left(\sum\limits_{j=0}^n{b_j\left(\sum\limits_{k=0}^p{c_kx^k}\right)^j}\right)^i}$ , associative.

4. matrix multiplication is associative.

Answer 2

Function composition is always associative. Matrices are really linear transformations on a vector space, so that fact should give you that both (3) and (4) are associative.

You will need to show (1) is associative, and for (2) you should be able to find a counterexample.