to draw a conclusion from a collection of premises, you need to construct a logical argument. A logical argument requires you to string together true statements that are dependent on each other, allowing you to walk your way through the problem and prove you know what you’re doing all at once. You’ll put these statements together in a truth table. Truth tables can involve as many statements as you like, but remember: it doesn’t matter what the statements actually say; what’s important is whether they’re true or false. Here’s how you build a truth table.
Note: to keep things simple, the following uses letters P and Q to represent statements.
If you’re in a negative mood and you have a statement hanging around, you might want to negate it as well. If you negate the statement P, you write it as ~P (and read it “not P”). So if P stands for the statement, “I like dogs,” ~P stands for “I don’t like dogs.”
Remember that all statements can be classified as either true or false. If P happens to be true, then ~P will be false. If P is false, on the other hand, then ~P will be true. P and ~P are always opposites of each other.
We’ll build our first truth table showing the relationship between P and ~P. The first column in a truth table is the statement P; the second column is the statement ~P. Now, P has only two options available: it can be either true or false (written in shorthand, T or F). I’ll put each option as a separate row in the following truth table. The first row in the truth table shows that if P is true, then ~P is false. The second row shows that if P is false, then ~P is true.
Individual statements can be combined to form compound statements. The statement “P and Q” is called the conjunction of P and Q and is written P ∧ Q. In order for the conjunction to be true, both P and Q have to be true. If either statement is false, the conjunction is false.
In order to construct a truth table for the compound statement P ∧ Q, we need to explore all possible combinations of truth values for both P and Q. When P is true, Q may be true or false. Similarly, when P is false, Q may be true or false. So the truth table will have four rows, one for each combination of truth values for P and Q.
|P||Q||P ∧ Q|
Example 2: Let P be the statement “Fire is hot” and Q be the statement “3 + 4 < 5.” Is P ∧ Q true or false?
Solution: Because Q is false, P ∧ Q is false. It doesn’t matter whether P is true or false. All it takes is one of the statements to be false to make the compound statement false.
A compound statement of the form “P or Q” is called the disjunction of P and Q. In symbols, the disjunction is written P ∨ Q. The only way that P ∨ Q is false is if both P and Q are false. On the other hand, all it takes for P ∨ Q to be true is for either P or Q to be true.
You can create even more complicated compound statements by combining “and” and “or.” In some cases, you can use parentheses to clarify the meaning of a compound statement. The information in the parentheses is given priority, just as it would be in an algebraic expression.
to build a truth table for a complicated compound statement, just pick apart the statement piece by piece. Start with P and Q, and then build the statements inside parentheses and work your way out. You can see the logical progression in the columns of the truth table for Example 3.
Example 3: Construct a truth table for the compound statement P ∧ (~P ∨ Q).
Solution: Begin with your truth values for P and Q, and then start building. First build ~P ∨ Q. to do this you’ll need a ~P column. After you build ~P ∨ Q you can combine it with P to finally determine the truth values for the compound statement P ∧ (~P ∨ Q).
|P||Q||~P||~P ∨ Q||P ∧ (~P ∨ Q)|
The final compound statement to consider is of the form “If P, then Q.” This statement is called an implication or a conditional statement. In symbols, you write P → Q (and read it “if P, then Q”). P is called the hypothesis (the given information) and Q is called the conclusion (it’s what you want to prove). The conditional statement makes a promise, which is broken if P is true and Q is false.
Consider the statement “If you clean your room, I promise to give you some ice cream.” What could happen here …?
Note that the only time the truth value of P → Q is false is if I break my promise: the statement P is true (the condition is met) and the statement Q is false (no follow-through). The truth table for P → Q is shown below.
|P||Q||P → Q|
Example 4: Construct a truth table for the compound statement (P → Q) ∨ P.
Solution: As usual, we start with columns for P and Q. Then we pick apart the compound statement, starting with the parentheses. We need a column for P → Q. When that’s done, we can determine the truth values for (P → Q) ∨ P.
|P||Q||P → Q||(P → Q) ∨ P|
It matters which statement is the hypothesis and which is the conclusion. Take the room-cleaning/ice-cream scenario. P → Q is quite different from Q → P. The ice cream is the reward; the cleaning of the room is not.
Even though P and Q aren’t interchangeable, you can rearrange them. Q → P is called the converse of the statement P → Q. If you’re in a particularly contrary mood, you might want to negate both statements. The statement ~P → ~Q is the inverse of the statement P → Q. Meanwhile, ~Q → ~P is called the contrapositive of the statement P → Q.
Example 5: If P is the statement “It’s raining” and Q is the statement “The sky is cloudy,” write the conditional (P → Q), the converse, the inverse, and the contrapositive compound statements using words, not symbols.
Solution: The conditional statement: “If it is raining, then the sky is cloudy.” The converse: “If the sky is cloudy, then it’s raining.” The inverse: “If it’s not raining, then the sky is not cloudy.” The contrapositive: “If the sky is not cloudy, then it’s not raining.”
Truth tables make discussing logical arguments much easier—and make Geometry much easier to understand! Good luck!
From The Complete Idiot’s Guide to Geometry, Second Edition, by Denise Szecsei, Ph.D.