Logic

Download A First Course in Mathematical Logic and Set Theory by Michael L. O'Leary PDF

By Michael L. O'Leary

A mathematical advent to the idea and purposes of common sense and set idea with an emphasis on writing proofs

Highlighting the purposes and notations of easy mathematical options in the framework of common sense and set conception, A First direction in Mathematical common sense and Set Theory introduces how common sense is used to arrange and constitution proofs and resolve extra advanced problems.

The e-book starts with propositional common sense, together with two-column proofs and fact desk purposes, by way of first-order good judgment, which gives the constitution for writing mathematical proofs. Set conception is then brought and serves because the foundation for outlining kin, features, numbers, mathematical induction, ordinals, and cardinals. The publication concludes with a primer on uncomplicated version conception with purposes to summary algebra. A First path in Mathematical good judgment and Set concept also includes:

  • Section workouts designed to teach the interactions among subject matters and make stronger the awarded principles and concepts
  • Numerous examples that illustrate theorems and hire simple techniques comparable to Euclid’s lemma, the Fibonacci series, and precise factorization
  • Coverage of significant theorems together with the well-ordering theorem, completeness theorem, compactness theorem, in addition to the theorems of Löwenheim–Skolem, Burali-Forti, Hartogs, Cantor–Schröder–Bernstein, and König

An very good textbook for college kids learning the rules of arithmetic and mathematical proofs, A First path in Mathematical good judgment and Set conception is additionally applicable for readers getting ready for careers in arithmetic schooling or machine technological know-how. furthermore, the ebook is perfect for introductory classes on mathematical common sense and/or set concept and applicable for upper-undergraduate transition classes with rigorous mathematical reasoning regarding algebra, quantity thought, or analysis.

 

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Additional info for A First Course in Mathematical Logic and Set Theory

Example text

In both the cases, the new inference rules will be justified using the rules we already know. 3. This motivates the next definition. 1 Let ???? and ???? be propositional forms. 10 For example, ???? , ????, ¬???? ∨ (???? → ????) ⊢∗ ???? because 1. 2. 3. 4. 5. 6. ???? ???? ¬???? ∨ (???? → ????) ???? → (???? → ????) ????→???? ???? Given Given Given 3 Impl 1, 4 MP 2, 5 MP is a formal proof using MP and Impl as the only inference rules. 10 except for MP. 2 For all propositional forms ???? and ????, ???? ⊢∗ ???? if and only if ???? ⊢ ????. PROOF Trivially, ???? ⊢∗ ???? implies ???? ⊢ ????, so suppose that ???? ⊢ ????.

We can use this to translate ???? ∧ (???? ∨ ????): the sine function is not one-to-one, and either the square root function is one-to-one or the absolute value function is not onto. 10 Chapter 1 PROPOSITIONAL LOGIC Be careful to note that the either-or phrasing is logically inclusive. For instance, some colleges require their students to take either logic or mathematics. This choice is meant to be exclusive in the sense that only one is needed for graduation. However, it is not logically exclusive. A student can take logic to satisfy the requirement yet still take a math class.

3. 2 INFERENCE 4. 5. 6. 7. 8. 19 Prove: ???? → ????, ???? → ????, ¬???? ⊢ ¬???? ∨ ¬???? 1. 2. 3. 4. 5. 6. ???? →???? ????→???? ¬???? (???? → ????) ∧ (???? → ????) ¬???? ∨ ¬???? ¬???? ∨ ¬???? Given Given Given 1, 2 Conj 3 Add 4, 5 DD Exercises 1. Show using truth tables. (a) ¬???? ∨ ????, ¬???? ⊨ ¬???? (b) ¬(???? ∧ ????), ???? ⊨ ¬???? (c) ???? → ????, ???? ⊨ ???? ∨ ???? (d) ???? → ????, ???? → ????, ???? ⊨ ???? (e) ???? ∨ ???? ∧ ????, ¬???? ⊨ ???? 2. Show the following using truth tables. (a) ¬(???? ∧ ????) ̸⊨ ¬???? (b) ???? → ???? ∨ ????, ???? ̸⊨ ???? (c) ???? ∧ ???? → ???? ̸⊨ ???? → ???? (d) (???? → ????) ∨ (???? → ????), ???? ∨ ???? ̸⊨ ???? ∨ ???? (e) ¬(???? ∧ ????) ∨ ????, ???? ∧ ???? ∨ ???? ̸⊨ ???? ∧ ???? (f) ???? ∨ ????, ???? ∨ ????, ???? ↔ ???? ̸⊨ ???? ∧ ???? 3.

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