How to prove the argument's validity using rules of inference.

Question

If I am late for my interview, then I will not get the job. I will get the job if I interview well. I will not interview well if I am late. If I catch a taxi, I will not be late for my interview. I got the job. Therefore I caught a taxi

I know it is invalid, but how to prove and provide a counter example?

Have a look at what I did.

Answer 1

Correct : in order to show the invalidtiy, you have to provide a counter example.

A counterexample is a truth assignment $v$ to the elementary sentences that satisfies all the premises and falsifies the conclusion.

Elementary sentences :

$L =$ I am late for my interview.

$J =$ I will get the job.

$W =$ I interview well.

$T =$ I catch a taxi.

Premises:

1) $L \to \lnot J$

2) $W \to J$

3) $L \to \lnot W$

4) $T \to \lnot L$

5) $J$

Conclusion :

6) $T$.

Thus, assume $v(T)=$ False.

Obviously, we want $v(J)=$ True, because $J$ is a premise.

With $v(T)=$ F we have also $v(T \to \lnot L)=$ T and with $v(J)=$ T we have $v(W \to J)=$ T.

Thus, up to now, we have satisfied : 2), 4) and 5).

Having $v(J)=$ T we can define $v(L)=$ F and thus also 1) is satisfied.

We are left with 3) $L \to \lnot W$. But we have $v(L)=$ F and thus also 1) is satisfied.

Conclusion : the truth assignment $v$ such that :

$v(L)=v(T)=$ F and $v(J)=$ T

is the required counter example.