Question

1. Answer the questions for the following argument.

If my laptop does not work, then I will use my sister’s computer.

My laptop works.

I will not use my sister’s computer.

1. [5 pts] Define statement variables (e.g., p, q) and formally rewrite the above argument into an argument form

2. [5 pts] Is the above argument form you answered valid or invalid? Why? Do not use a truth table when justifying your answer.

Answer 1

Let p : My laptop works
q : I will use my sister’s computer

a) If my laptop does not work, then I will use my sister’s computer.

Answer: ¬p  $\rightarrow$ q

My laptop works.

Answer: p

I will not use my sister’s computer.

Answer: ¬q

b) Given (¬p  $\rightarrow$ q) $\wedge$ (p) $\wedge$ (¬q)

$\equiv$ (¬p  $\rightarrow$ q) $\wedge$ (p) $\wedge$ (¬q)

$\equiv$ (¬(¬p)  V q) $\wedge$ (p) $\wedge$ (¬q)   { Law of Implies (P$\rightarrow$Q) = ¬PVQ }

$\equiv$ (p  V q) $\wedge$ (p) $\wedge$ (¬q)  {By De Morgan's law ¬ (¬P) = P }

$\equiv$ (p  V p) $\wedge$ (q) $\wedge$ (¬q) {Commutative law (P $\wedge$ Q) = (Q $\wedge$ P)}

$\equiv$ (p V p)  $\wedge$ (q $\wedge$ ¬q) ​​​​​​​  {Associative law (P Q) R = P (Q R)}

$\equiv$ (p) $\wedge$ (F) ​​​​​​​   { we know that P $\wedge$ ¬P = F & P V P = P }

$\equiv$ (p $\wedge$ F) ​​​​​​​ {Associative law (P Q) R = P (Q R)}

$\equiv$  (F) ​​​​​​​ { we know that P $\wedge$ F = F }

$\equiv$  False

So the above argument is invalid