MAD2104/Final Exam Page 8 of 817. (10 pts.) The following is a valid argument:"Each of five friends, Larry, Moe, Curly, Samantha, andAnitra, has taken a course in discrete mathematics and passedwith a grade of at least a C. Every student who has taken andpassed a discrete math class with a grade of at least a C cantake a course in algorithms. Therefore, all five friends cantake a course in algorithms."The validity of the argument can be seen easily bysymbolizing the argument using propositional functions andquantifiers as follows:Define propositional functions as follows:f(x) : "x is one of the friends listed."d(x) : "x has taken and passed discrete math with at least a C."a(x) : "x can take a course in algorithms."Then the argument translates into this:(∀x)(f(x)→d(x))(∀x)(d(x)→a(x))∴(∀x)(f(x)→a(x))In the proof of validity which follows provide thejustification(s) for each step. In doing this, you shouldexplicitly cite rules of inference and quantification,hypotheses, and preceding steps by number.1.(∀x)(f(x)→d(x))Justification:2.f(y)→d(y)Justification:3.(∀x)(d(x)→a(x))Justification:4.d(y)→a(y)Justification:5.f(y)→a(y)Justification:6.(∀x)(f(x)→a(x))Justification:
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