Binary Operation Introduction

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Problem 39 (Binary Operation Introduction)

In which of the following scenarios do we have a binary operation on a set $G$. Justify your answers.

1. Addition $a+b$ of integers.
2. Multiplication $A\cdot B$ of 3 by 3 matrices.
3. The dot product $\vec u\cdot \vec v$ of two dimensional vectors.
4. The cross product $\vec u\times \vec v$ of three dimensional vectors.
5. The scalar product $c\cdot \vec v$.
6. Composition $f\circ g$ of permutations on the same set $X$.
7. Composition $f\circ g$ of functions from $X$ to $Y$.

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