Math 213 Supplement (3rd ed rev) (Fall 2011)pp 10 to 11

Math 213 Supplement (3rd ed rev) (Fall 2011)pp 10 to 11 -...

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Logic Before we add further to our list of assumptions and theorems, it is important to examine logic itself a bit more closely. In math, our reasoning is more strict than in everyday language. When a mother tells her child, “If you don’t behave, you won’t get any ice cream!” the child feels it is fair to assume that if he does behave, he will get ice cream. However, strictly speaking, the mother didn’t promise him any ice cream for good behavior, she only promised no ice cream for bad behavior. A different example may illustrate this more clearly. Let’s assume the following statement is true: “If it is nighttime, then you can’t see the sun.” In our part of the world this is true. Now, the following are all variations of the statement. Which are equivalent to the original statement? Converse : If you can’t see the sun, then it is nighttime. Inverse : If it is not nighttime, then you can see the sun. Contrapositive : If you can see the sun, then it is not nighttime. Some days it’s cloudy – on these days you can’t see the sun, but it’s not nighttime. This contradicts two of the variations, so the converse and inverse above are not equivalent to the
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Math 213 Supplement (3rd ed rev) (Fall 2011)pp 10 to 11 -...

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