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While we've seen that it's possible for a statement to be true while its converse is false, it turns out that the contrapositive is better behaved. Written in English, the inverse is, "If it is not a mirror, then it is not shiny," while the contrapositive is, "If it is not shiny, then it is not a mirror." If we abbreviate the first statement as mirror → shiny, then the inverse would be not mirror → not shiny and the contrapositive would be not shiny → not mirror. What is the inverse of the statement "All mirrors are shiny?" What is its contrapositive? Again in symbols, the contrapositive of p → q is the statement not q → not p, or ~ q → ~ p. So we can also write the inverse of p → q as ~ p → ~ q.įinally, if you negate everything and flip p and q (taking the inverse of the converse, if you're fond of wordplay) then you get the contrapositive. Sometimes mathematicians like to be even more brief than this, so they'll abbreviate "not" with the symbol "~". For example, if you negate (that means stick a "not" in front of) both the hypothesis and conclusion, you get the inverse: in symbols, not p → not q is the inverse of p → q. There are some other special ways of modifying implications.
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We want to switch the hypothesis and the conclusion, which will give us: "If something has seeds, then it is a watermelon." Of course, this converse is obviously false, since apples, cucumbers, and sunflowers all have seeds and are not watermelons. Write the converse of the statement, "If something is a watermelon, then it has seeds." To highlight this distinction, mathematicians have given a special name to the statement q → p: it's called the converse of p → q. It's kind of like subtraction: 5 – 3 gives a different answer than 3 – 5. In other words, p → q and q → p mean very different things. The hypothesis and conclusion play very different roles in conditional statements. In (B), we may rewrite the statement as "If I peel rutabagas, then I cut off a finger," telling us that p = "I peel rutabagas" and q = "I cut off a finger."įinally, we may rewrite (C) as "If it is a dog, then it will go to heaven," yielding p = "it is a dog" and q = "it will go to heaven." (B) I cut off a finger whenever I peel rutabagas.įor (A), p = "it rains outside" and q = "flowers will grow tomorrow." (A) If it rains outside, then flowers will grow tomorrow. Identify p and q in the following statements, translating them into p → q form. In fact, the old saying, "Mind your p's and q's," has its origins in this sort of mathematical logic. However, mathematicians can be drier than the Sahara desert: they tend to write conditional statements as a formula p → q, where p is the hypothesis and q the conclusion.
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Some ways to mix it up are: "All things satisfying hypothesis are conclusion" and " Conclusion whenever hypothesis." The same is true of conditional statements: after a while, the If-Then formula becomes a real snoozefest. As your English teacher would say, good writers vary their sentence structure.