Determine whether the following arguments is valid or not. If valid, explain how

Question

Determine whether the following arguments is valid or not. If valid, explain how the conclusion can be derived from the premises. For each step of the derivation provide proper justification and mention the interface fule you are using, if any. If not valid, explain why. Begin by identifying the atomic propositions involved and then translating each of the statements into propositional logic. No need to use predicate logic for this question.

(a) She is a Math Major or Computer Science Major.

(b) If she does not know discrete math, she is not a Math Major.

(c) If she knows discrete math, she is smart.

(d) She is not a Computer Science Major.

(e) Therefore, she is smart.

Explanation / Answer

Atomic Prepostitions Involved :
so lets say :
M : she is maths major
C : she is computer scienc emajor
D : She know desrete maths .
S : she is smart

so here rules say : Tanslating them to Propositional Logic

(a) She is a Math Major or Computer Science Major.
C1 : M OR C

(b) If she does not know discrete math, she is not a Math Major.
C2 : ! D -> ! M

(c) If she knows discrete math, she is smart.
C3 : D -> S

(d) She is not a Computer Science Major.
C4 : ! C

so from , a & d                            the interface fule
M OR C
! C
____________________
:- M                                          (by disjunctive syllogism ) ----- (1)

from b and above (1)
! D -> ! M
M
____________________
:- D                                   ( MODES TOLLENS ) -------------(2)

from c and (2)
D
D -> S
____________________
:- S             she is smart                           (Modes Ponens)

So we reduced to (e) ,i.e. she is smart .

Hence this is a valid argument

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