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# Logic and Proof

# 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 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.