A proposition is a simple declarative sentence that can be classified (by everyone) as either true or false, but not both. As we combine propositions with operations we get statements perhaps not so simple but that have truth values. We designate variable propositions by the letters p, q, and r.
Ex. 1.
Let p be the statement "2 + 4 = 5," and let q be the statement "UNCW is in Wilmington."
p is certainly false, while q is true.
We could write: "UNCW is in Wilmington and 2 + 4 ¹ 5." as "q and not p".
When propositions are combined to form more complex statements we have a statement form.
The simplest operation that can be applied to a statement is negation (Ø ). If q is true, then Ø q is false. If q is false, then Ø q is true.
The next simplest operation is the conjunction (Ù ). This binary operator has the meaning "p and q" and will be true only when both p and q are true.
The disjunction operator (Ú ) has the meaning "p or q" and will be true when either p or q is true. "p or q" is false only when both propositions are false. Consider the statement, "The waiter offers sugar or cream with your coffee." Getting both sugar and cream is not excluded. The general meaning of Ú is inclusive.
Ex. 2.
Write each of the following sentences symbolically, letting p = "It is hot" and q = "It is sunny."
(a) It is not hot but it is sunny.
(b) It is neither hot nor sunny.
The convention in logic is that but and and mean the same thing.
The phrase neither A nor B means the same as not A and not B.
Ex. 3.
Construct a truth table for p Ù Ø q. (The unary operator Ø takes precedence over the binary operator Ù .)
|
p |
q |
p Ù Ø q |
|
T |
T |
T F F |
|
T |
F |
T T T |
|
F |
T |
F F F |
|
F |
F |
F F T |
The solution is given by the middle truth value in the final column. Writing of the first and third values is merely an optional aid. Be sure to underline the solution column if this aid is utilized.
The exclusive or ( Å
) has the meaning "p x-or q" and will be true when either p or q is true. However,
"p Å q" is false when both propositions are false or when both propositions are true. Consider the statement, "This object is a motorcycle or it is a book." The object can be either, but it can not be both.
The essential point about assigning truth values to compound statements is that it allows you -using logic alone- to judge the truth of a compound statement based on your knowledge of the truth of its component parts. Logic does not help you determine truth of the component statements.
TruthTable.java / Truth Tables for not, conjunction, disjunction, exclusive or, and the conditional operators.