prove that the arguments are valid using rules of inference and laws of predicate logic, (state the laws/rules used)
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a)
1)
2)
From 1) we can get:-
3) [Universal instantiation using an element 'a' within the domain of x]
From 2) we get:-
4) [Existential instantiation using an element 'a' within the domain of x]
From 4) we get:-
5) P(a) [Simplification]
From 3) and 5) we get:-
6) [Modus Ponens]
From 6) we get:-
7) S(a) [Simplification]
From 4) we get:-
8) R(a) [Simplification]
From 7) and 8) we get:-
9) [Conjunction]
From 9) we get:-
10) [Existential generalization]
Hence Proved.
b)
1)
2)
3)
4)
From 1) we get:-
5) P(a) V Q(a) [Universal instantiation using an element 'a' within the domain of x]
From 2) we get:-
6) ~Q(a) V S(a) [Universal instantiation using an element 'a' within the domain of x]
From 3) we get:-
7) R(a) -> ~S(x) [Universal instantiation using an element 'a' within the domain of x]
From 4) we get:-
8) ~P(a) [Existential instantiation using an element 'a' within the domain of x]
From 7) we get:-
9) S(a) -> ~R(a) [Contrapositive]
From 5) and 6) we get:-
10) P(a) V S(a) [Resolution]
From 10) and 8) we get:-
11) S(a) [Disjunctive syllogism]
From 9) and 11) we get:-
12) ~R(x) [Modus Ponens]
From 12) we get:-
13) [Existential generalization]
Hence Proved.
The solutions of the given problems as follows;
prove that the arguments are valid using rules of inference and laws of predicate logic, (state...
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