Solutions To Mathematics Textbooks/Proofs and Fundamentals/Chapter 7

=Section 7.1=

7.1.2
(2) The binary operation * in $$ \mathbb{R} $$ defined by $$  x * y = x + 2y $$  $$\forall x,y\in \mathbb{R} $$

Answer: The binary operation * in $$\mathbb{R} $$ is not commutative, counter example: With $$ 2,3\in \mathbb{R} $$: $$ 2 * 3 = 2 + 2 (3) = 8 $$ and $$ 3 * 2 = 3 + 2 (2) = 7 $$ Like $$ 7 \neq 8 $$, the definition of commutativity is not satisfied.

=Section 7.2=