Consider the statement "Suppose that it's raining. Then there is a cloud in the sky.".
(i)Determine the hypotheses/assumptions and the conclusion.(ii)Rewrite this statement explicitly in the form "If A, then B" using Part (i).(iii)Is this statement true or false?
Solution.(i)The hypothesis we are making is that it is raining. The conclusion we are making is that there must be a cloud in the sky.(ii)"If it's raining, then there must be a cloud in the sky."(iii)This statement is true. (Based on all that is currently known about how rain works!)Consider the statement "If x is a positive integer or a solution to x+3>4, then x>0 and x>12." Is this statement true?Solution. To determine if it's true, let's look first at the assumptions. We are assuming that either x is a positive integer, or that it solves the inequality x+3>4.
Next let's consider the conclusion. We are concluding that x must satisfy both inequalities x>0 and x>12. If we look more closely, we see that once we satisfy the second inequality, the first is redundant. (If x>12, then it must already be larger than zero.)
Now, in order for this statement to be true, we need that if x solves either of the assumptions, then it must solve x>12. Well, the first assumption is that x is a positive integer, which means that x≥1, so in this case the conclusion holds. The second assumption is that x+3>4, or equivalently, that x>1, which means the conclusion holds as well.