This question already has an answer here: An infinite subset of a countable set is countable 2 answers

Still struggling with proofs. This formula was presented as a given in my book and it wasn’t intuitive to me at all, so I wanted to verify it as it seems images and inverse images play important roles in the material that follows. Even the examples I constructed didn’t give me an intuitive understanding. $X, […]

Consider the following sets: $$ A = \{1, 2, \{1,2\}, \emptyset \} $$ $$ B = \emptyset $$ My book says that $|A| = 4$ and $|B| = 0$. Why is $\emptyset$ considered an element if it’s a subset, but not when it’s on its own?

Prove or disprove: If $A\times B\sim A\times C$, then $B\sim C$. (“$\sim$”: “numerically equivalent” / “has the same cardinality as”) What bijection/counterexample should I use to prove/disprove it?

For every positive number there exists a corresponding negative number. Would that imply that the number of positive numbers is “equal” to the number of negative numbers? (Are they incomparable because they both approach infinity?)

Problem statement: Let A be an uncountable set, B a countable subset of A, and C the complement of B in A. Prove that there exists a one-to-one correspondence between A and C. My thoughts: There’s a bijection between A and A (the identity function). There’s a bijection between C and C (the identity function). […]

What kind of mathematical object is this substitution(is it a function or what). We assuming set of variables exist.

I tried doing this by defining a function $f$ which takes an element (a subset of $\mathbb{N}$) and maps it to an infinite sequence of $0$s & $1$s. A subset, i.e the element of $P(\mathbb{N})$ will contain discrete numbers from $\mathbb{N}$ (arranged in increasing order, since the order of numbers in a subset does not […]

I am trying to solve an exercize, in which we were asked to prove that Zorn’s lemma implies axiom of choic. I am using a guidness that was given that said we should use a set $\mathcal{F}$ which i’ll define throughout the proof: Proof: Given a set of nonempty sets $F$, define the set $\mathcal{F}$ […]

How many different subsets of a $10$-element set are there where the subsets have at most than $9$ elements? I know there are $2^{10}$ total number of $10$-element sets. Please explain to me this problem.

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