1. ## Inference and Quantifiers

Explain which rules of inference are used for each step.

"Each of 5 roommates, Melissa, Aaron, Ralph, Veeneesha, & Keeshawn, has taken a course in discrete mathematics. Every student who has taken a course in discrete mathematics can take a course in algorithms. Therefore, all 5 roommates can take a course in algorithms next year."

2. Here is the inference in symbolic form in the case of Melissa.

$\displaystyle \begin{array}{lll} 1 & \mathsf{TakenDiscreteMathematics}(\mathsf{Melissa} ) & \text{Assumption}\\ 2 & \forall x\,\mathsf{TakenDiscreteMathematics}(x)\to \mathsf{CanTakeAlgorithms}(x) & \text{Assumption}\\ 3 & \mathsf{TakenDiscreteMathematics}(\mathsf{Melissa} )\to \mathsf{CanTakeAlgorithms}(\mathsf{Melissa}) & \text{From 2}\\ 4 & \mathsf{CanTakeAlgorithms}(\mathsf{Melissa}) & \text{From 3, 1} \end{array}$

It remains for you to name the rules used in steps 3 and 4.

In fact, assumption #1 may have been $\displaystyle \mathsf{TakenDiscreteMathematics}(\mathsf{Melissa} )\land\mathsf{TakenDiscreteMathematics}(\mathsf{Aa ron})$$\displaystyle {}\land\mathsf{TakenDiscreteMathematics}(\mathsf{R alph})\land\mathsf{TakenDiscreteMathematics}(\math sf{Veeneesha})$$\displaystyle {}\land\mathsf{TakenDiscreteMathematics}(\mathsf{K eeshawn})$. Then one has to break this first into individual conjuncts, conclude $\displaystyle \mathsf{CanTakeAlgorithms}$ for each student as shown above for Melissa, and then assemble those facts into another conjunction.