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

ISBN-10: 0495562025

ISBN-13: 9780495562023

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**Example text**

Working forward from x2 ≤ 1, we have −1 ≤ x ≤ 1, so x ≤ 1. Therefore, x < 5 and x < 2, from which we can conclude that x − 5 < 0 and x − 2 < 0, which is exactly what we need. Proof. Assume that x 2 ≤ 1. Then −1 ≤ x ≤ 1. Therefore x ≤ 1. Thus x < 5 and x < 2, and so we have x − 5 < 0 and x − 2 < 0. Therefore, (x − 5)(x − 2) > 0. Thus x2 − 7x + 10 > 0. Hence x2 − 7x > −10. Ⅲ We now consider direct proofs of statements of the form P ⇒ Q when either P or Q is itself a compound proposition. We have in fact already constructed proofs of statements of the form (P ∧ Q) ⇒ R.

Statement (b) is true because the truth set of x 2 = 0 is precisely {0} and thus is nonempty. Since the open sentence x 2 = −1 is never true for real numbers, the truth set of x 2 = −1 is empty. Statement (c) is false. In the universe ގ, only statement (a) is true. The three statements are all true in the universe {0, 5, i} and all three statements are false in the universe {1, 2}. Sometimes we can say (E x) P (x) is true even when we do not know a specific object in the universe in the truth set of P(x), only that there (at least) is one.

Is indeed a special case of the quantifier E . ” The proofs are left to Exercise 11. 3 ଁ 1. Translate the following English sentences into symbolic sentences with quantifiers. The universe for each is given in parentheses. ૺ (a) Not all precious stones are beautiful. (All stones) ଁ (b) All precious stones are not beautiful. (All stones) (c) Some isosceles triangle is a right triangle. (All triangles) (d) No right triangle is isosceles. (All triangles) (e) All people are honest or no one is honest.

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

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