Chapter 3.2, Problem 12E

Assign a grade of A (correct), C (partially correct), or F (failure) to each. Justify assignments of grades other than A.

1. Claim. Let o be the operation on $ℝ\times ℝ$ defined by setting $\left(a,b\right)\circ \left(c,d\right)=\left(a+c,b+d\right),$ and let - be the usual subtraction on $ℝ.$ Then the function f given by $f\left(a,b\right)=a-3b$ is an OP map from $\left(ℝ\times ℝ,\circ \right)$ to $\left(ℝ,-\right).$
   "Proof." $\left(4,2\right)$ and $\left(3,1\right)$ are in $ℝ\times ℝ.$ Then $f\left(\left(4,2\right)\circ \left(3,1\right)\right)=f\left(7,3\right)=7-3\cdot 3=-2,$ whereas $f\left(4,2\right)-f\left(3,1\right)=-2-0=-2,$ sof is operation preserving.

Claim. Let $f:\left(G,*\right)\to \left(H,\cdot \right)$ and $g:\left(H,\cdot \right)\to \left(K,\otimes \right)$ be OP maps. Then the composite $g\circ f:$ $\left(G,*\right)\to \left(K,\otimes \right)$ is an OP map.
"Proof." $g\circ f\left(ab\right)=g\left(f\left(ab\right)\right)=g\left(f\left(a\right)f\left(b\right)\right)=g\left(f\left(a\right)\right)g\left(f\left(b\right)\right)=\left(g\circ f\left(a\right)\right)\left(g\circ f\left(b\right)\right).$