(a) Let P(x,y) be a boolean function. Assume that VyP(x, y) is Truc and that the domain of discourse is nonempty. Which of the following must also be true? If the statement is true, explain your answer; otherwise,give a counter-example. \text { i. } \forall x \exists y \neg P(x, y) \text { ii. } \forall x \forall y P(x, y) \text { iii. }-x=\exists y r^{>}(x, y) -) Given the following argument: "The football game will be cancelled only If it rains "it rained, therefore the football game was cancelled" Assume p means " it rains" whereas q means "football game cancelled" i. Translate this argument to a symbolic form. ii. Construct the truth table. iii. Determine if this argument is a valid argument or not. Say whether or not the following argument is a valid argument. Explain your answer. (a) Successful candidates for this job must have either a Master's degree or five years of work experience (b) Johnny has a Master's degree (c)Johnny got the job (d) Johnny does not have five years of work experience d) Let P(x) and Q(z) be two predicates and suppose D is the the domain ofx. For the statement forms in each pair, determine whether they have the same truth value for every choice of p(x), Q(x) and D, or not. \text { i. } \forall x \in L,(P(x) \wedge Q(x)) \text {, and }\left(\forall x \in L, P^{\prime}(x)\right) \wedge(\forall x \in D, Q(x)) \text { ii. } \forall x \in D,(P(x) \vee Q(x)) \text {, and }(\forall x \in D, P(x)) \vee(\forall x \in D, Q(x))

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