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Write "All citizens of Egypt speak Arabic," in p → q form.
First, we can rewrite this statement into If-Then form and then translate it into p → q form. The statement would be, "If someone is a citizen of Egypt, then he/she speaks Arabic." That means being a citizen of Egypt is our hypothesis and speaking Arabic is our conclusion. In p → q form, Egyptian citizen → Arabic speaker.
Suppose the converse of a statement is "If there are clouds in the sky, then it is raining." What is the original statement?
The converse of a statement is switching the hypothesis and the conclusion. All we have to do, then, is change the positions of "there are clouds in the sky" and "it is raining." It doesn't take a crane to do that. Our original statement becomes, "If it is raining, then there are clouds in the sky."
What is the contrapositive of "Any American president must be born in America"? Is it true?
Since If-Then statements are the basis, we'll stick with that and rewrite it, "If you are an American president, then you are born in America." In that case our p → q formula takes the form American president → born in America. So far, so good.
The contrapositive is the inverse of the converse. First, we switch 'em. That means born in America → American president. Then the inverse means putting "not" in front of both of them. That translates to not born in America → not American president.
Time to abandon the Mathspeak. The contrapositive then states, "If you are not born in America, then you are not an American president." This statement will always be true no matter how much Arnold Schwarzenegger wishes it weren't.