Logic and Sets for AMC-8

Statements and their converses

Let us represent logical statements by capital letters. The statement “If A then B” is a conditional statement involving the statements A and B.

Example: If there is a storm, then the party shall be cancelled.

Converse : If B, then A.

Inverse : If not A, then not B.

Contrapositive : If not B, then not A.

A statement is logically equivalent to its contrapositive. Clearly, the inverse is the contrapositive of the converse.

Sets

Sets are well-defined collections of objects. There are three common ways to describe a set. Let us give an example

In words : S is the set of natural numbers less than or equal to n.

List notation : S=\{1,2,\cdots n\}.

Set builder notation : S=\{i\in\mathbb{N}|1\le i\le n\}.

Visual representation by Venn diagrams

X\cup Y :=\{z|z\in X\;\text{or}\;z\in Y\}

X\cap Y :=\{z|z\in X\;\text{and}\;z\in Y\}

Relations and Mapping

Cartesian product of two sets C and D is defined as C\times D :=\{(c,d)|c\in C,d\in D\}. A relation between C and D is a subset R of C\times D. If (a,b)\in R then a is said to be related to b. We denote this statement by a\sim b. The relation R is said to be a function from C to D if

(i) Given any x\in C, there is a y\in D such that (c,d)\in R.

(ii) (a,b)\in R and (a,c)\in R implies that b=c.

The function is said to be

(i) injective if a\sim b and c\sim b implies a=c.

(ii) surjective if given any d\in D there exists a c\in C such that c\sim d.

(iii) bijective if it is both injective and surjective.

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