Articles of elementary set theory

Is every subset of $\mathbb{Z^+}$ countable?

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

Prove the formula $ff^{-1}(B) = B \cap f(X) \subset B$ where $f: X\to Y$

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, […]

Why does the empty set have a cardinality of zero?

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?

Proving: If $|A\times B| = |A\times C|$, then $|B|=|C|$.

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?

Are there an equal number of positive and negative numbers?

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?)

Proof there is a 1-1 correspondence between an uncountable set and itself minus a countable part of it

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). […]

substituting a variable in a formula (in logic)

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

Prove Cardinality of Power set of $\mathbb{N}$, i.e $P(\mathbb{N})$ and set of infinite sequences of $0$s & $1$s is equal.

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 […]

Proving that Zorn's Lemma implies the axiom of choice

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 $9$ elements?

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.