proposition Logic

Question

Please explain 

Comments

draw kmap and simplify. Enjoy :)

Answer 1

Your choice of option (B) means you know (2) and (3) are Valid.

A statement is Valid if it always gives true value, irrespective of inputs. and we know that $p \Rightarrow q$ is equal to $\overline{p} + q$

Option-1: $(p \Rightarrow q) \land (r \Rightarrow s) \land (p \lor q) \Rightarrow (q \lor s)$

$(\overline{p} + q)(\overline{r} + s)(p + q) \Rightarrow (q + s)$ (represented in boolean form)

$\overline{(\overline{p} + q)(\overline{r} + s)(p + q)} + q + s$

$\overline{(\overline{p} + q)} + \overline{(\overline{r} + s)} + \overline{p+q} + q + s$

$p.\overline{q} + r.\overline{s} + (\overline{p}.\overline{q} + q)  + s$

$p.\overline{q} + r.\overline{s} + q + \overline{p} + s$

$(p.\overline{q} + q) + r.\overline{s} + \overline{p} + s$

$p + q + r.\overline{s} + \overline{p} + s$

$(p + \overline{p}) + q + r.\overline{s}  + s$

$1 +q + r.\overline{s}  + s$ which is equal to $1$(true) and is thus Valid.

Thus all options are Valid.