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

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Which of the following are true? ૺ (a) Triangles have three sides iff squares have four sides. (b) 7 + 5 = 12 iff 1 + 1 = 2. ૺ (c) b is even iff b + 1 is odd. ) (d) m is odd iff m2 is odd. ) (e) 5 + 6 = 6 + 5 iff 7 + 1 = 10. (f) A parallelogram has three sides iff 27 is prime. (g) The Eiffel Tower is in Paris if and only if the chemical symbol for helium is√H. √ √ √ √ √ √ √ 10 + 13 < 11 + 12 iff 13 − 12 < 11 − 10. (h) (i) x 2 ≥ 0 iff x ≥ 0. ) (j) x 2 − y 2 = 0 iff (x − y)(x + y) = 0. ) (k) x 2 + y 2 = 50 iff (x + y) 2 = 50.

F) (Ex)(3(2 − x) = 5 + 8(1 − x)). ()ޒ (g) (∀x)(x 2 + 6x + 5 ≥ 0). ()ޒ (h) (∀x)(x 2 + 4x + 5 ≥ 0). ()ޒ (i) (Ex)(x 2 − x + 41 is prime). ()ގ (j) (∀x)(x 2 − x + 41 is prime). ()ގ (k) (∀x)(x 3 + 17x 2 + 6x + 100 ≥ 0). ()ޒ (l) (∀x)(∀y)[x < y ⇒ (Ew)(x < w < y)]. ()ޑ 9. Give an English translation for each. The universe is given in parentheses. (a) (∀x)(x ≥ 1). x)(x ≥ 0 ∧ x ≤ 0). ()ޒ (c) (∀x)(x is prime ∧ x = 2 ⇒ x is odd). x)( loge x = 1). ()ޒ Copyright 2011 Cengage Learning, Inc.

X)(x ≥ 0 ∧ x ≤ 0). ()ޒ (c) (∀x)(x is prime ∧ x = 2 ⇒ x is odd). x)( loge x = 1). ()ޒ Copyright 2011 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 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)].