Use the logical equivalence established in Example, to rewrite the following statement. (A...

Use the logical equivalence established in Example, to rewrite the following statement. (Assume that x represents a fixed real number.)

If x > 2 or x<–2, then x2 > 4.

Example

Division into Cases: Showing that p ∨ q → r ≡ ( p → r) ∧ (q → r)

Use truth tables to show the logical equivalence of the statement forms  and . Annotate the table with a sentence of explanation.

Solution

First fill in the eight possible combinations of truth values for p, q, and r . Then fill in the columns for p ∨ q, p →r , and q →r using the definitions of or and if-then. For instance, the p →r column has F’s in the second and fourth rows because these are the rows in which p is true and is false. Next fill in the p ∨ q →r column using the definition of if-then. The rows in which the hypothesis p ∨ q is true and the conclusion r is false are the second, fourth, and sixth. So F’s go in these rows and T’s in all the others. The complete table shows that p ∨q →r and (p →r ) ∧ (q →r ) have the same truth values for each combination of truth values of p, q, and r . Hence the two statement forms are logically equivalent