Discrete mathematics veii imp

Question

Prove the validity of the following argument " If I get the job and work hard,then I'll get promoted.

If I get promoted then i'll be happy.

I will not be happy.

Therefore either i will not get the job or i will not work hard."

Answer 1

let p: I get the job
q: I work hard
r: I'll get promoted

p^q-->r for the first argument

now

t:I'll be happy
so for 2nd argument it becomes r->t

now we have 3rd argument as ~t

so from modus tollens

r-->t
~t
therefore ~r

Now combining arguments

 p^q->r  and ~r

from modus tollens

, we get ~(p^q) = ~p OR ~q
hence proved either I will not get the job or I will not work hard