Here's the solution to the problem: The binary operation $\oplus$ is defined as $p \oplus q = \frac{pq}{7} + p + q$. a) Inverse of -2 under $\oplus$ Step 1: Find the identity element $e$. The identity element $e$ satisfies $p \oplus e = p$ for any real number $p$. $$ p \oplus e = \frac{pe}{7} + p + e $$ Setting this equal to $p$: $$ \frac{pe}{7} + p + e = p $$ Subtract $p$ from both sides: $$ \frac{pe}{7} + e = 0 $$ Factor out $e$: $$ e \left( \frac{p}{7} + 1 \right) = 0 $$ $$ e \left( \frac{p+7}{7} \right) = 0 $$ For this equation to hold for all $p$ (except $p=-7$), the identity element must be $e=0$. Step 2: Find the inverse of -2. Let the inverse of -2 be $x$. By definition, $(-2) \oplus x = e$. Since $e=0$: $$ (-2) \oplus x = 0 $$ Using the definition of the operation: $$ \frac{(-2)x}{7} + (-2) + x = 0 $$ $$ -\frac{2x}{7} - 2 + x = 0 $$ Multiply the entire equation by 7 to eliminate the fraction: $$ -2x - 14 + 7x = 0 $$ Combine like terms: $$ 5x - 14 = 0 $$ Add 14 to both sides: $$ 5x = 14 $$ Divide by 5: $$ x = \frac{14}{5} $$ The inverse of -2 is $\boxed{\frac{14}{5}}$. b) Truth set of $m \oplus 5 = m \oplus 7$ Step 1: Expand $m \oplus 5$. Using the definition $p \oplus q = \frac{pq}{7} + p + q$, with $p=m$ and $q=5$: $$ m \oplus 5 = \frac{m \cdot 5}{7} + m + 5 = \frac{5m}{7} + m + 5 $$ Step 2: Expand $m \oplus 7$. Using the definition $p \oplus q = \frac{pq}{7} + p + q$, with $p=m$ and $q=7$: $$ m \oplus 7 = \frac{m \cdot 7}{7} + m + 7 = \frac{7m}{7} + m + 7 = m + m + 7 = 2m + 7 $$ Step 3: Set the two expressions equal and solve for $m$. $$ \frac{5m}{7} + m + 5 = 2m + 7 $$ Combine the $m$ terms on the left side: $$ \frac{5m}{7} + \frac{7m}{7} + 5 = 2m + 7 $$ $$ \frac{12m}{7} + 5 = 2m + 7 $$ Multiply the entire equation by 7 to eliminate the fraction: $$ 12m + 35 = 14m + 49 $$ Subtract $12m$ from both sides: $$ 35 = 2m + 49 $$ Subtract 49 from both sides: $$ 35 - 49 = 2m $$ $$ -14 = 2m $$ Divide by 2: $$ m = \frac{-14}{2} $$ $$ m = -7 $$ The truth set is $\boxed{\{-7\}}$.