$($Answer must be in your own hand$-$writing$)$

$($a$)$ Using truth table method, show that the hypotheses $"$If it rains today, then we will not have barbeque today." and $"$If we do not have barbeque today, then we will have it tomorrow." imply the conclusion $"$If it rains today, then wee will have barbeque tomorrow $"$

$(2$ marks$)$

$($b$)$ Let $p,q,r$ and $s$ be the statements 'The weather is warm', The sky is clear', $\text{'}$We go swimming' and $\text{'}$We go boating' respectively. Symbolize the following argument and prove its validity by using inference theory:

Il the sky is clear, then either we go swimming of we go borting. Tho sley: is clear and we do not go swimming. Therefore, we go boating," $(3$ marks$)$