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Logic and Proof

Logic and Proof


The other main way of pasting together statements is by using the word "or," forming a disjunction. The statement "A or B" is true exactly when at least one of A or B is true. If both happen to be true, then "A or B" remains true. This is another way mathematical language differs from common speech.

When we snidely say, "I can take out the trash or I can wash the dishes," we usually mean we'll do one or the other, but not both. Of course, if you have a mathematician parent, they'll likely say, "True, and you'll do both."

Sample Problem

Let's say x = 4 and y = 17. Is the statement, "x is prime or y is a perfect square" true?

Again, we'll split the statement into atoms and figure out the truth of each atom. We first ask whether x is prime. Well, 4 has a factor of 2, so this part is definitely false. There's still hope though! Is y a perfect square? Four squared is 16 and five squared is 25, which skips over y. This statement gets a big fat "False" as well. Since neither part is true, we're forced to conclude that the whole statement is false.

To summarize:

  • Not (or negation): "not A" is true when A is false, and is false when A is true.
  • And (or conjunction): "A and B" is true when both A is true and B is true, and is false when either is false.
  • Or (or disjunction): "A or B" is true when either A or B (or both!) is true, and is false when both are false.
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