1.3 Predicates and Quantifiers

DEFINITIONS:

①     Predicates is a logical operator that returns a Boolean value that is either true or false.

We also use predicates in programming.

Consider the statement “x is absent in the class today.”

Ø  Since x is an unknown, this statement is not a propositional statement.

However, we do know that when we assign a value (object) into x, the whole statement will make sense as it explains how a subject (x) is being affected.

Hence, we can say that x is a subject which can take any value into the statement to turn the whole statement into a propositional statement and it can hold a set of truth values.

Likewise, “is absent in the class today” represents what does the property that x holds.

Ø  Hence, “is absent in the class today” is known as the predicate for x (subject).

Ø  The predicate and subject, if combined together, can be written as P(x) where P is the predicate and x is the subject of the statement. It is also known as the propositional function.

In general, a statement of the form P(X_1, X_2, X_{3, …,} X_n) is the value of proposition function P at n-tuple (X_1, X_2, X_{3, …,} X_n) and P is known as n-place predicate or n-ary predicate.

1.2 Propositional Equivalences

DEFINITION:

A compound proposition that is always true, no matters what the truth values of the propositional variable that occur in it, is called TAUTOLOGY. A compound proposition that is always false is CONTRADICTION neither is called CONTINGENCY.

LOGICAL EQUIVALENCES

Compound propositions that have the same truth values in all possible cases are called logically equivalences .

                                                     @

The compound propositions p and q are called logically equivalence if p↔q is a tautology. The notation p≡q denotes that p and q are logically equivalent.

REMARK : The sym ≡is not a logical connective and p ≡ q is not a compound proposition but rather is the statement that p↔q is a tautology. The symbol ó is used instead of ≡ to denote logical equivalent.

1.1 Proposition Logic

Chap 1: The Foundations- Logics and Proofs

1.1 Proposition Logic

DEFINITIONS:

①     Propositional is a declarative sentence that is either true or false, but not both. It normally uses lower case roman letters (p, q, r,…) to denote its propositional variables (statement variables), denoted by T when the statement is true and F for false statement.

②     Logic means the study of either of proposition logic or first-order predicate logic.

p/s: Logic encompasses the following attributes:

l   Syntax         : Define the syntactically acceptable object of language. Also

known as formulae

l  Semantics    : Formulae of logic associate each formula with a meaning.

l  Proof-theory : Manipulating formulae according to the perceived reasoning.

③     Atomic proposition is a truth or falsity that does not depend on the truth or falsity of any other proposition.

Discrete Mathematic- An Preview

"As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."

Albert Einstein (1879 - 1955), "Geometry and Experience", January 27, 1921

     Discrete mathematics is the study structural mathematics fundamentally focusing in the study of objects which are finite and infinite. Therefore, study of continuous mathematics such as “calculus” and “algebra” are different from the study of discrete mathematics. Often in time, the structures behind a mathematical theorem and logics will be elaborated in this course, and students are required to define or even construct a new theorem based on the learned topics. Thus, mathematical reasoning is prioritized so as to read, understand, and define mathematical arguments and statements using methods of proof. These concepts are often useful in studying and solving problems related to computer science and logic, such as automata theory, cryptography, software testing, software development and programming languages using proposed algorithms based on the proven theories.

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