Recent content by trixitium

I'm reading the first chapter of Topology by Munkres. There we can see: "if f is bijective, there exists a function from B to A called the inverse of f . (...) As another situation where care is needed, we note that it is not in general true that f^{-1}(f(A_0) = A_0 and...

Yes, C' is the complement of C if x \in A \cap B , by the definition of intersection: A \cap B = \{x \in A : x \in B\} and we can conclude that x is simultaneously in A and B. But my doubt, is how to reduced the (A \cap C) \cup (B \cap C') to a expression that i can...

Homework Statement Show that: A \cap B \subset (A \cap C) \cup (B \cap C') Homework Equations The Attempt at a Solution I tryed distribute (A \cap C) over (B \cap C') but I'm always walking in circles and i don't came to a satisfactory answer. This exercise was in a section...

Homework Statement Determine if the following set is a vector space under the the given operations. The set V of all pairs of real numbers of the form (1,x) with the operations: (1, y) + (1, y') = (1, y + y') k(1, y) = (1, ky) Homework Equations The Attempt at a Solution...

It also fails in: K is any scalar, u is in V, ku is in V. u = (x,y) If I choose k < 0, then ku = k(x,y) = (kx,ky) and kx < 0 and ku is not in V.

Homework Statement Determine if the following set is a vector space under the given operations. List all the axioms that fail to hold. The set of all pairs of real numbers of the form (x,y), where x >= 0, with the standard operations on R^2 Homework Equations The Attempt at a...

Hello, I would like to solve this exercise in the best way as possible. I solved using the most trivial way and I am in doubt if are there some better way to solve. Homework Statement Simplify: Homework Equations \left(x + \frac{1}{x} \right)\left(y + \frac{1}{y} \right) +...