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

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

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The truth set of the open sentence “x2 < 5” depends upon the collection of objects we choose for the universe of discourse. With the universe specified as the set ‫ގ‬, the truth set is {1, 2}. For the universe ‫ޚ‬, the truth set {−2, −1, 0, 1, 2}. √ is √ When the universe is ‫ޒ‬, the truth set is the open interval (− 5, 5). Copyright 2011 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 3 Quantifiers 19 DEFINITION With a universe specified, two open sentences P(x) and Q(x) are equivalent iff they have the same truth set.

We also write P iff Q to abbreviate P if and only if Q. The truth table for P ⇐ ⇒ Q is P Q P⇐ ⇒Q T F T F T T F F T F F T Examples. The proposition “23 = 8 iff 49 is a perfect √ square” is true because both components are true. The proposition “π = 22/7 iff 2 is a rational number” is true because both components are false. The proposition “6 + 1 = 7 iff Lake Michigan is in Kansas” is false because the truth values of the components differ. Definitions, fully stated with the “if and only if” connective, are important examples of biconditional sentences because they describe exactly the condition(s) to meet the definition.

B) P ⇐ ⇒ P ∧ ∼Q. (c) P ⇒ Q ⇐ ૺ (d) P ⇒ [P ⇒ ( P ⇒ Q)]. ⇒ P. (e) P ∧ (Q ∨ ∼Q) ⇐ (f) [Q ∧ (P ⇒ Q)] ⇒ P. ⇒ Q) ⇐ ⇒ ∼(∼P ∨ Q) ∨ (∼P ∧ Q). (g) (P ⇐ (h) [P ⇒ (Q ∨ R)] ⇒ [(Q ⇒ R) ∨ (R ⇒ P)]. ⇒ Q) ∧ ∼Q. (i) P ∧ (P ⇐ (j) (P ∨ Q) ⇒ Q ⇒ P. (k) [P ⇒ (Q ∧ R)] ⇒ [R ⇒ ( P ⇒ Q)]. (l) [P ⇒ (Q ∧ R)] ⇒ R ⇒ (P ⇒ Q). 17. The inverse, or opposite, of the conditional sentence P ⇒ Q is ∼P ⇒ ∼Q. (a) Show that P ⇒ Q and its inverse are not equivalent forms. (b) For what values of the propositions P and Q are P ⇒ Q and its inverse both true?

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