Unique Things

Lecture# 01 Operation Research: What is Operation Research? Operation Research(O.R) is an art of winning wars without actual fighting. (Arthur Clark) O.R is a scientific approach to the problems. (H.M.…

Real Valued Function Let \(f: A \to \mathbb{R}\) be a function. Clearly domain of \(f\) is \(A\), in other words \(f\) is defined on \(A\). Since co-domain of \(f\) is…

The function \(f(x) = x^{(x - 1)}\) is not: (A) Uniform continuous on \((0,1)\) (B) Continuous on \((0,1)\) (C) Differentiable on \([0,1)\) (D) Both A and B \(\text{Ln } z\)…

Observations on Vector Spaces \(F \in V\) → possible chances of vector space \(V \in F\) → not a vector space Following this observation: \(\mathbb{Q}(\mathbb{Q})\) vector space (Q is a…

Recommended Books: E. Rees, Notes on Geometry, Springer, 2004. M. A. Armstrong, Groups and Symmetry, Springer, 1998. H. Eves, Fundamentals of Modern Elementary Geometry, Jones and Bartlett Publishers International, 1992…

Gauss (1777- 1855) a German mathematician was first to prove in a satisfactory manner that some algebraic equations with real coefficients have complex roots in the form \(x\pm iy\) (Note…

Partial Contents Rings, commutative ring, ring with unity (identity), examples Consequences from the definition Unit element of ring Division ring or skew field, field, zero divisor Integral domain Characteristic of…