By Daniel W. Cunningham

The ebook is meant for college kids who are looking to find out how to end up theorems and be larger ready for the trials required in additional strengthen arithmetic. one of many key elements during this textbook is the improvement of a technique to put naked the constitution underpinning the development of an explanation, a lot as diagramming a sentence lays naked its grammatical constitution. Diagramming an explanation is a manner of offering the relationships among some of the elements of an evidence. an explanation diagram presents a device for exhibiting scholars find out how to write right mathematical proofs.

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Additional info for A Logical Introduction to Proof

Example text

These connectives are also used to tie together a variety of mathematical statements. A good understanding of these logical connectives will allow us to more easily understand and construct mathematical proofs. 2 1. 2. 3. 4. Using truth tables, show that ¬(P → Q) ⇔ (P ∧ ¬Q). Construct truth tables to show that (P ↔ Q) ⇔ (P → Q) ∧ (Q → P). Using truth tables, show that P ⇔ (¬P → (Q ∧ ¬Q)). Which of the following statements are true and which are false? (a) (b) (c) (d) (e) (π 2 > 9) → (π > 3). If 3 ≥ 2, then 3 ≥ 1.

Show that the following argument is invalid: (a) If x ≥ 2, then x ≥ 0. (b) x ≥ 0. Therefore, x ≥ 2. Solution. Assertion (a) is a true statement. Let x = 1. Thus (b) is also true, while the conclusion is false. So the argument is invalid. Example 6 (Converse Error). Show that the argument P→Q Q ∴P is invalid. 3 Valid and Invalid Arguments 23 Solution. Assign the truth values of P, Q to be F, T respectively. With this truth assignment we see that the conclusion is false while all of the premises are true.

Thus, the statement D(9, 27) is true and D(9, 5) is false. The domain of a predicate is just the collection of allowed values for the variable(s) in the predicate. So, the domain of the predicate T (x) is the collection of all people. 3). Example 1. Consider the three predicates P(x), E(x), and D(x, y) given by • P(x) symbolizes the statement “x is a prime number” • E(x) symbolizes the statement “x is even” • D(x, y) symbolizes the statement “x evenly divides y” where x and y represent integers.