Symbolize the following arguments using the letters provided, and (b) show that each argument is valid by constructing proofs

Symbolize the following arguments using the letters provided, and (b) show that each argument is valid by constructing proofs of their conclusions from their premises using the rules of inference we have studied in this course. (12/50)
(a) If Smith once beat the fireman at billiards, then Smith is not the fireman. Smith once beat the fireman at billiards. If the brakeman is Jones, then Jones is not the fireman. The brakeman is Jones.
If Smith is not the fireman and Jones is not the fireman, then Robinson is the fireman. If the brakeman is Jones and Robinson is the fireman, then Smith is the engineer. Therefore, Smith is the engineer. (O— Smith once beat the fireman at billiards; M— Smith is the fireman; B— the brakeman is Jones; N— Jones is the fireman; F— Robinson is the fireman; G— Smith is the engineer.)
(b) If Xavier takes Spanish, then Yolanda tutors him. Zeke pays Yolanda if she tutors Xavier. Either Waldo or Xavier takes Spanish. Waldo does not take Spanish; also Yolanda doesn’t tutor Xavier.
So, Zeke pays Yolanda, but Waldo doesn’t take Spanish. (X— Xavier takes Spanish; Y— Yolanda tutors Xavier; Z— Zeke pay Yolanda; W— Waldo takes Spanish.)
(c) If Anderson was nominated, then she went to Boston. If she went to Boston, then she campaigned there. If she campaigned there, she met Douglas. Anderson did not meet Douglas. Either Anderson was nominated or someone more eligible was selected. Therefore, someone more eligible was selected. (A— Anderson was nominated; B— Anderson went to Boston; C— Anderson campaigned in Boston; D— Anderson met Douglas; E— Someone more eligible was selected.)
(d) If Santacruz is injured, then she will not play. And if Santacruz does not play, then Khumalo must play. What’s is more, if Khumalo and Nakata both play, then Jones must be moved to the center. In fact, Santacruz is injured and Nakata will play. Thus, Khumalo will play Jones will be moved into the center. (I— Santacruz is injured; S— Santacruz will play; K— Khumalo must play; N— Nakata plays; J— Jones must be moved to the center.)

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