Wednesday April 2
"or" in English means exclusive or, "or" in mathematics means inclusive.
"implies" in English suggests a logical connection, where "A => B" only means "it never occurs that A is true and B is false". So e.g. if B is always true, then "A => B" is true as a mathematical statement, even if they seem completely unrelated. Same if A is always false!
"is" in English often doesn't translate as =, but as "is an element of". So "Felix is a cat" doesn't mean any cat = Felix, but that Felix is an element of {the set of cats}.
Sets are determined by their elements. Subsets, power sets, union, intersection, cardinality.
Definition of an empty set. Theorem: there is only one empty set (meaning, any two such are equal). Proof: the only thing we know about sets is that they are equal if they have the same elements, which is true for any two empty sets. QED.
Next time: graphs.
"implies" in English suggests a logical connection, where "A => B" only means "it never occurs that A is true and B is false". So e.g. if B is always true, then "A => B" is true as a mathematical statement, even if they seem completely unrelated. Same if A is always false!
"is" in English often doesn't translate as =, but as "is an element of". So "Felix is a cat" doesn't mean any cat = Felix, but that Felix is an element of {the set of cats}.
Sets are determined by their elements. Subsets, power sets, union, intersection, cardinality.
Definition of an empty set. Theorem: there is only one empty set (meaning, any two such are equal). Proof: the only thing we know about sets is that they are equal if they have the same elements, which is true for any two empty sets. QED.
Next time: graphs.
posted by Allen Knutson @ 1:55 PM
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