Fundamentals of Logic

Fundamentals of Logic

To make complicated mathematical relationships clear,it is convenient to use the notation of symbolic logic.Symbolic logic is about statements which one can meaningfully claim to be true or false and there is no statement can be both true and false.

Definition

Any statement A has a negation ¬A defined by ¬A is true if A is false, and ¬A is false if A is true.

Two statements, A and B, can be combined using conjunction '∧' and disjunction '∨' to make newstatements. The statement A∧B is true if both A and B are true, and is false in all other cases. The statement A∨B is false when both A and B are false, and it is true, if A is true, if B is true, or if both A and B are true.

A⇒B:=(¬A)∨B.Thus A⇒B is false if A is true and B is false.A⇒B is true when A and B are both true, or when A is false.This means that a true statement can't imply a false statement, and also that a false statement implies any statement—true or false. What's more, A⇒B also means that A is a sufficient condition for B, and B is a necessary condition for A.

The equivalence A⇐⇒B the statements is defined by A⇐⇒B:=(A⇒B)∧(B⇒A),which can be think as:A is true if and only if B is true.

The statement ¬B⇒¬A is called the contrapositive of the statement A⇒B.And we have A⇒B is true if and only if ¬B⇒¬A is true.

In mathematics，a true statement is often called a proposition, theorem, lemma or corollary.

If x belongs to a class(or set) X, that is,x is an element of X, then we write x∈X.