If p and q are true and r and s are false statements, then which of the following is true?

Options

A. `(q ^^ r) vv (~p ^^s)`
B. `(~ p to q) hArr (r ^^s)`
C. `(p to q) vv (r hArrs)`
D. `(p^^~r) ^^(~p vv s)`

Answer

C

Explanation

We have statements `p, p to T` and `r,s to F`
Option (a) `( q ^^ r) vv(-p ^^s) = (T ^^F) v(F^^ F) = FvvF = F`
Option (b) `(~ p to q) hArr ( r^^s) -= (p vvq) to (r vv s) (because p to q = ~ p vv q)`
` -= (~ p vv q) vv(r ^^s) ^^((p vvq) vv (~ r ^^ s)))(because p hArr q -= (~ pvvq) ^^ (p vv -q)`
` -=(~ (TvvT) vv(F ^^F) ^^((T vvF) vv(F ^^ F))`
` -=(F vv F) ^^(T vvT) -= F ^^ T -= F`
Option (c) `(p to q) vv(r hArr s)`
` -= (~pvv q) vv((~ r vvs) ^^( rvv -s))(because p to q =- p vv q)`
` = (FvvT) vv((T vvF) ^^(F vvT))`
` = T vv (T ^^ T) -= T vv T = T`
Option (d) do similar as option (a).