Fundamentals of Math

Fundamentals of Mathematics

8.1 Truth Tables To help us keep track of our assessment in an orderly fashion, mathematicians, like yourself, will often use a truth table. A truth table is a diagram in rows and columns showing how the truth or falsity of a proposition varies with that of its components. Let's look at the truth tables for our unary operator first.

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Cell 1

Cell 2

T

F

Cell 3

Cell 4

F

T

Cells 1 and 3 tell us the truthfulness of the original sentence; in cell 1, the sentence is true, and cell 3, the sentence is false. By negating the sentences in columns 1 and 3, we see the truthfulness changes. The negated true sentence becomes false (cell 3), and the negated false sentence becomes true (cell 4). Let's look at an example; assume the sentence states, "Every triangle has three sides." We can evaluate that sentence and determine that it is true. If we negate the sentence, "Not every triangle has three sides," then we must re-evaluate it. When we do that, we see the sentence becomes false, just as predicted by the truth table.

Now, let's review the truth tables for the binary operators we discussed earlier. Remember, and represent independent sentences, respectively.

Conjunction (and)

Conditional (If … then …)

∧ T F F F

→

T T F F

T F T F

T T F F

T F T F

T F T T

Disjunction (or)

Biconditional (… if and only if …) ↔ T T T T F F F T F F F T

∨ T T T F

T T F F

T F T F

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