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I'm reading Naive Set Theory by Paul H. Halmos and there's something that has been bugging me for a while. In the book, Halmos seems to assume $2^A=\mathscr P(A)$, and I don't quite understand the rea …

By definition, we have $|A|\le|B|$ iff there exists an injection between $A$ and $B$. Furthermore, if $|A|\le|B|$ and $|B|\le|A|$, then $|A|=|B|$ (that is, there exists a bijection between $A$ and $B$ …

The notation usually used to denote the power set of a set $A$ is basically the same as that used to denote a function having $A$ as its argument: $\mathcal P(A)$. However, is it actually a function? …

Today I had a discussion with my teacher (I'm in high school). She said that you can verify if a function $f$ is surjective by finding its inverse $f^{-1}$ and verifying that its domain equals the cod …

Instead of beginning from the natural numbers, we first define $\mathbb R$ using field axioms. Let $\mathscr H$ be a set of subsets of $\mathbb R$ defined as follows: $$\mathscr H = \{H \subset \mathb …

Let me clarify my question. I know what the domain of a function is, formally. However, it is common to see exercises such as Determine the domain of the function $f(x)=\dfrac{\sqrt{x^2\log\sqrt[3]{x …

This is a merely formal question. I'll explain with an example: say I want to denote the set of all the real numbers which have a reciprocal greater than $1$. I would write it like this: $$S = \left\{ …

I know that, as sets, $\mathbb C$ and $\mathbb R^2$ are exactly the same set, and that the difference is about the structure: $\mathbb C$ is $\mathbb R^2$ with a field structure. Does this mean that, …