Computational Logic and Discrete Structure: Set Theory Definition: Set A well-defined collection of distinct objects is called a set. e.g. {1,2,3,4,5}, {a,b,c} etc. Note: 1) Set is always denoted by A, B, C etc. 2) Elements of a set are denoted by a, b, c, x, y, z etc. 3) If 'a' is an element of set ‘A’ then we write $a\in\ A$, otherwise, $a\notin A$.

Computational Logic and Discrete Structure: Set Theory Definition: Set A well-defined collection of distinct objects is called a set. e.g. {1,2,3,4,5}, {a,b,c} etc. Note: 1) Set is always denoted by A, B, C etc. 2) Elements of a set are denoted by a, b, c, x, y, z etc. 3) If 'a' is an element of set ‘A’ then we write $a\in\ A$, otherwise, $a\notin A$.

If ​\( F'(x)=f(x) \)​ then F is called primitive or antiderivative of f. e.g.\( F(x)=x^2sin(\frac{1}{x}) \) ​\( \therefore F'(x)=2xsin(\frac{1}{x})+x^2.cos(\frac{1}{x}).(\frac{-1}{x^2}) \)​ ​ $\therefore F'(x)=2xsin(\frac{1}{x})-cos(\frac{1}{x})$ ​ Here, ​ $F'(0)=\displaystyle\lim_{x\to0}\frac{F(x)-F(0)}{x-0}$ ​                        ​ $=\displaystyle\lim_{x\to0}\frac{x^2sin(\frac{1}{x})}{x}$ ​                       ​$ =\displaystyle\lim_{x\to0}xsin(\frac{1}{x})$ ​                       ​ $=0 ​$

Let (X, d) be a metric space and suppose that it is disconnected. Claim: ​\( \exists \)​ a non-empty proper subset of X which is both open & closed. Since, X is disconnected by definition, ​\( \exists \)​ non-empty sets A & B such that ​\( X=A\cup B, \bar{A}\cap B=\phi, \bar{B}\cap A=\phi \)​.