# A transition to advanced mathematics by Smith D., Eggen M., Andre R.

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

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X) P (x)? (a) (∀x)P(x) ∨ (∀x) ∼ P(x). (b) (∀x) ∼P(x) ∨ (Ey)(Ez)(y = z ∧ P( y) ∧ P(z)). (c) (∀x)[P(x) ⇒ ( Ey)(P(y) ∧ x = y)]. ૺ (d) ∼ ( ∀x)(∀y)[(P(x) ∧ P( y)) ⇒ x = y]. 4 14. Riddle: What is the English translation of the symbolic statement ∀E E ∀? Basic Proof Methods I In mathematics, a theorem is a statement that describes a pattern or relationship among quantities or structures and a proof is a justification of the truth of a theorem. Before beginning to examine valid proof techniques it is recommended that you review the comments about proofs and the definitions in the Preface to the Student.

May not be copied, scanned, or duplicated, in whole or in part. qxd 18 CHAPTER 1 4/22/10 1:42 AM Page 18 Logic and Proofs 15. Give the converse and contrapositive of each sentence of Exercises 10(a), (b), (c), and (d). Tell whether each converse and contrapositive is true or false. 16. Determine whether each of the following is a tautology, a contradiction, or neither. ૺ (a) [(P ⇒ Q) ⇒ P] ⇒ P. ⇒ P ∧ (P ∨ Q). (b) P ⇐ ⇒ P ∧ ∼Q. (c) P ⇒ Q ⇐ ૺ (d) P ⇒ [P ⇒ ( P ⇒ Q)]. ⇒ P. (e) P ∧ (Q ∨ ∼Q) ⇐ (f) [Q ∧ (P ⇒ Q)] ⇒ P.

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.