By Smith D., Eggen M., Andre R.

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**Example text**

If a > 5, then a > 3. a > 5 implies a > 3. a > 5 is sufficient for a > 3. a > 5 only if a > 3. a > 3, if a > 5. a > 3 whenever a > 5. a > 3 is necessary for a > 5. a > 3, when a > 5. ⇒ Q to translate: Use P ⇐ Examples: P if and only if Q. P if, but only if, Q. P is equivalent to Q. P is necessary and sufficient for Q. |t | |t| |t| |t| = 2 if and only if t2 = 4. = 2 if, but only if, t 2 = 4. = 2 is equivalent to t 2 = 4. = 2 is necessary and sufficient for t 2 = 4. The word unless is one of those connective words in English that poses special problems because it has so many different interpretations.

You now have an equation to use. Second, if you need to prove that x is odd, then the definition gives you something equivalent to work toward: It suffices to show that x can be expressed as x = 2k + 1, for some natural number k. You’ll find it useful in writing proofs to keep in mind these two ways we use definitions. Example. ” In this example, the equivalence of the two statements comes from our knowledge of the real numbers that (ab = 0) ⇐ ⇒ (a = 0 or b = 0). Tautologies are important both because a statement that has the form of a tautology may be used as a step in a proof, and because tautologies are used to create rules for making deductions in a proof.

4 10. ૺ ૺ ૺ ૺ 11. ଁ ଁ Basic Proof Methods I 27 (e) ∼ (Ex)(x2 < 0). x)(x2 = 0). ()ޒ (g) (∀x)(x is odd ⇒ x2 is odd). ()ގ Which of the following are true in the universe of all real numbers? (a) (∀x)(Ey)(x + y = 0). (b) (Ex)(∀y)(x + y = 0). (c) (Ex)(Ey)(x2 + y2 = −1). (d) (∀x)[x > 0 ⇒ ( Ey)(y < 0 ∧ xy > 0)]. (e) (∀y)(Ex)(∀z)(xy = xz). (f) (Ex)(∀y)(x ≤ y). (g) (∀y)(Ex)(x ≤ y). y)(y < 0 ∧ y + 3 > 0). x)(∀y)(x = y2). x)(x = y2). y)(∀w)(w2 > x − y). Let A(x) be an open sentence with variable x. 2 (a).