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Math Help - Inference and Quantifiers

  1. #1
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    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."
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  2. #2
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    Here is the inference in symbolic form in the case of Melissa.

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

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

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