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Beginner
2 hrs 55 mins
12 chapters
Overview
Mathematical logic is one of the fundamental topics in discrete mathematics. In fact, it's used in building the foundation of all mathematics, in general.
It is the science of reasoning, intuition; all done symbolically. Having its roots back in the time of the ancient Greeks, mathematical logic has evolved enormously over the years.
What we know today is a very sophisticated, formal system to work with reasoning.
Understanding mathematical logic is of immense importance if you ought to understand mathematical proofs — how to use given knowledge to derive new things; how to switch between conditionals and their contrapositives to simplify proof work; and so on.
Prerequisites
No special prerequisites for this course.
Learning outcomes
- The history of mathematical logic — where it all began; which logicians played a big role in the formalisation of logic and so on.
- What is propositional calculus, also known as propositional logic; how to use it in expressing simple declarative sentences.
- What is the difference between atomic and compound propositions, and that how to simplify working with propositions using propositional variables.
- What are logical operators, also known as logical connectives, and how to use them in combining different propositions together into one single propositions. Moreover, you'll be able to relate the behavior of all logical operators with natural language.
- What is negation, disjunction, conjunction, implications and bi-implication, and how to represent each of these operations symbolically.
- What are truth tables, and how to use them to evaluate the truth value of a given proposition in all possible cases; to understand the semantics of given operators; to check whether given propositions are always true, or always false; and so on.
- What are valuation and interpretation functions, and how can they used to formally define propositional satisfiability and validity.
- The different variations of a given implication — inverse, converse and contrapositive — and that which variation conveys the same meaning as the original implication statement.
- The concept of propositional equivalences, which is vital in simplifying given propositions by removing long expressions with equivalent shorter expressions. You'll also know what are the two laws of De Morgan, along with many other simple laws.
- What is predicate calculus, also known as predicate logic, or as first-order logic.
- What are predicates and how are they the foundational concept of predicate calculus.Moreover, you'll know how to combine predicates with given subjects using propositional functions.
- What are quantifiers and how are they used extensively throughout mathematics to assert something regarding a group of objects. You will be familiar with the two most common quantifiers — universal quantifier and existential quantifier.
- How to combine, two or more quantifiers together to express many statements otherwise not capable of being expressed with single quantifiers.
Table of contents
In this unit, we'll discover the most fundamental area of logic, which is believed to be the foundation of logic — propositional calculus. We'll work with propositions, see how to combine them using logical operators, how to draw truth tables for given propositional expressions, how to work with implications, how to work with propositional equivalences and so on. We'll also discover one very famous set of equivalences named after the British mathematician, Augustus De Morgan.
In this unit, we shall unravel the ideas behind predicate calculus, which is the next step after propositional calculus. We'll discover what are predicates and how to combine them with given objects to create propositions. We'll then see what are quantifiers and how to use them to express a huge variety of statements in logic.
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