 # 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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