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Let $X$ be a set and $\sim$ is an equivalence relation on $X$, so that the quotient set $X/_\sim=\bigcup_{x\in X}{[x]}$ with $[x]=[y]$ if an...

I try to understand the actual intuition behind the logarithm properties and came across a post on this site that explains the multiplicati...

Let $A = \{a_1, a_2, . . . , a_n\}$ where $a_1 Show that $\phi(a_i) = a_i$ for $i = 1, 2,\ldots , n.$ We prove this by contradiction and ind...

In high school textbooks, the following characterizations are often found: A function is continuous if its graph can be drawn without lif...

Definition of smooth curve: A parametric curve $\mathrm Z(t)=x(t)+i > y(t)$ on $[a,b]$ is called smooth if $\mathrm Z'(t)=...

Image : http://postimg.org/image/dkn0d5uen/ I'm studying Spivak's calculus and I have a really simple question : I'm only in the...

If $x\neq0$ , how to find the reciprocal of $x$ only by using trigonometric functions and inverse trigonometric functions? I have found onl...