Question

In: Computer Science

2. Determine whether the following Propositional Logic statements are valid or invalid arguments. You may use...

2. Determine whether the following Propositional Logic statements are valid or invalid arguments. You may use a truth table, proofs using rules of inference, or resolution (specify which method you are using).

(a) p→q, ~q→r, r; ∴p

(b) ~p∨q, p→(r∧s), s→q; ∴q∨r

(c) p→(q→r), q; ∴p→r

Solutions

Expert Solution

a) Truth Table:

   p

q

r

~q

(p q)

(~q r)

      (p q)(~q r)( r)

   p

    0

    0

    0

    1

    1

    0

                          0

    0

    0

    0

    1

    1

    1

    1

                          1

    0

    0

    1

    0

    0

    1

    1

                          0

    0

    0

    1

    1

    0

    1

    1

                          1

    0

    1

    0

    0

    1

    0

    0

                          0

    1

    1

    0

    1

    1

    0

    1

                          0

    1

    1

    1

    0

    0

    1

    1

                          0

    1

    1

    1

    1

    0

    1

    1

                          1

    1

From the above truth table if we observe the last TWO column which are not Equal

So the given Argument is Invalid.

b)Truth Table:

    p

    q

    r

    s

    ~p

     r s  

    ~p V q

p(rs)

sq

   (~p V q) (p(rs))(sq)

    q V r

    0

    0

    0

    0

    1

    0

    1

    1

    1

    1

    0

    0

    0

    0

    1

    1

    0

    1

    1

    0

    0

    0

    0

    0

    1

    0

    1

    0

    1

    1

    1

    1

    1

    0

    0

    1

    1

    1

    1

    1

    1

    0

    0

    1

    0

    1

    0

    0

    1

    0

    1

    1

    1

    1

    1

    0

    1

    0

    1

    1

    0

    1

    1

    1

    1

    1

    0

    1

    1

    0

    1

    0

    1

    1

    1

    1

    1

    0

    1

    1

    1

    1

    1

    1

    1

    1

    1

    1

    1

    0

    0

    0

    0

    0

    0

    0

    1

    0

    0

    1

    0

    0

    1

    0

    0

    0

    0

    0

    0

    0

    1

    0

    1

    0

    0

    0

    0

    0

    1

    0

    1

    1

    0

    1

    1

    0

    1

    0

    1

    0

    0

    1

    1

    1

    0

    0

    0

    0

    1

    0

    1

    0

    1

    1

    1

    0

    1

    0

    0

    1

    0

    1

    0

    1

    1

    1

    1

    0

    0

    0

    1

    0

    1

    0

    1

    1

    1

    1

    1

    0

    1

    1

    1

    1

    1

    1

From the above truth table if we observe the last TWO column which are not Equal

So the given Argument is Invalid.

c) Truth Table:

   p

q

r

(q r)

      p (q r)

[p (q r)](q)

(p r)

    0

    0

    0

    1

    1

    0

    1

    0

    0

    1

    1

    1

    0

    1

    0

    1

    0

    0

    1

    1

    1

    0

    1

    1

    1

    1

    1

    1

    1

    0

    0

    1

    1

    0

    0

    1

    0

    1

    1

    1

    0

    1

    1

    1

    0

    0

    0

    0

    0

    1

    1

    1

    1

    1

    1

    1

From the above truth table if we observe the last TWO column which are not Equal

So the given Argument is Invalid.

venereology answered 9 months ago
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