Determine Whether the Given Argument Is Valid or Invalid C) Invalid

question 123

Multiple Choice

determine whether the given argument is valid or invalid. If it is invalid, select a counterexample.
-1. (?x) (?y) (Pxy • ?Pyx)
2) (?x) [(?y) Pxy ? (?y) Qxy] / (?x) (?y) (Qxy • ?Pyx)

• A) Valid
• B) Invalid. Counterexample in a domain of three members, in which:
• \(\begin{array}{lll}
• \text { Paa: True } & \text { Pab: True } & \text { Pac: False } \\
• \text { Pba: False } & \text { Pbb: True } & \text { Pbc: True } \\
• \text { Pca: True } & \text { Pcb: False } & \text { Pcc: True } \\
• \text { Qaa: False } & \text { Qab: False } & \text { Qac: True } \\
• \text { Qba: True } & \text { Qbb:True } & \text { Qbc: True } \\
• \text { Qca: True } & \text { Qcb:True } & \text { Qcc: True }
• \end{array}\)
• C) Invalid. Counterexample in a domain of three members, in which:
• \(\begin{array}{lll}
• \text { Paa: True } & \text { Pab: True } & \text { Pac: False } \\
• \text { Pba: False } & \text { Fbb: False } & \text { Pbc: False } \\
• \text { Pca: False } & \text { Pcb: False } & \text { Pcc: True } \\
• \text { Qaa: False } & \text { Qab: False } & \text { Qac: True } \\
• \text { Qba: True } & \text { Qbb:True } & \text { Qbc: False } \\
• \text { Qca: True } & \text { Qcb: True } & \text { Qcc: True }
• \end{array}\)
• D) Invalid. Counterexample in a domain of three members, in which:
• \(\begin{array}{lll}
• \text { Paa: False } & \text { Pab: Ture } & \text { Pac: False } \\
• \text { Pba: False } & \text { Pbb: True } & \text { Pbc: True } \\
• \text { Pca: True } & \text { Pcb: True } & \text { Pcc: True } \\
• \text { Qaa: False } & \text { Qab:True } & \text { Qac: True } \\
• \text { Qba: False } & \text { Qbb:True } & \text { Qbc: True } \\
• \text { Qca: True } & \text { Qcb:True } & \text { Qcc: False }
• \end{array}\)
• E) Invalid. Counterexample in a domain of three members, in which:
• \(\begin{array}{lll}
• \text { Paa: True } & \text { Pab: True } & \text { Pac: False } \\
• \text { Pla: False } & \text { Pbb: False } & \text { Ploc: True } \\
• \text { Pca: True } & \text { Pcb: False } & \text { Pcc: True } \\
• \text { Qaa: False } & \text { Qab: False } & \text { Qac: True } \\
• \text { Qba: False } & \text { Qbb: False } & \text { Qbc: False } \\
• \text { Qca: True } & \text { Qcb: True } & \text { Qcc: False }
• \end{array}\)

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