At a Glance - Disjunctions
The other main way of pasting together statements is by using the word "or," forming a disjunction. The statement "A or B" is true when either A is true, B is true, or both are 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."
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.
- 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.