$($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 we 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:

$"$If the sky is clear, then either we go swimming or we go boating. The sky is clear and we do not go swimming. Therefore, we go boating." $(3$ marks$)$