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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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 Spring '12
 mclaren

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