Articles of elementary set theory

Proving $\bigcup A – \bigcup B \subset \bigcup(A-B)$

I need to prove: $\bigcup\limits_{i} A – \bigcup\limits_j B \subset \bigcup\limits_j (A-B)$ $\bigcap\limits_{i} A – \bigcap\limits_j B \subset \bigcup\limits_j (A-B)$ So when can the equality hold? Appreciate your help.

Finding a bijective function from integer to words

I need to find a function to enumerate the ordered list of sequential words based on a charset. Let me give you an example. If the charset is “abc”, the function to be found “f” should compute the following: f(0) = a f(1) = b f(2) = c f(3) = aa f(4) = ab f(5) […]

Confused about why “disjointifying” implies “AC”

Assume I have the following (DIS) For every indexed family $\{A_i : i \in I \}$ there exists a family $\{B_i : i \in I \}$ of pairwise disjoint sets such that $B_i \subset A_i$ for all $i \in I$ and $\bigcup_{i \in I} B_i = \bigcup_{i \in I} A_i$. and (AC) For every family […]

Possible inaccuracy in Wikipedia article about initial ordinals

I quote from the Wikipedia article: “So (assuming the axiom of choice) we identify $\omega_\alpha$ with $\aleph_\alpha$, except that the notation $\aleph_\alpha$ is used for writing cardinals, and $\omega_\alpha$ for writing ordinals. “ Let’s remind ourselves of the definitions: (Def.1) The $\aleph_\alpha$ numbers are defined recursively, with $\aleph_0 = |\mathbb N| = \omega = \omega_0$ […]

Cartesian product of bijective functions is bijective

If $A, B, C, D$ are sets such that $A \sim B$ and $C \sim D$, $\exists$ bijections $f: A \to B$ and $g: C \to D$. Let $h: A \times C \to B \times D$ be $h(a,c) = (f(a), g(c))$. Show that $h$ is a bijection (and thus $A\times C \sim B \times D$). […]

List applications of sets & relations in science/business/tech that a highschooler can understand

What are some applications of sets & relations in science/business/tech that a highschooler can understand? To kindle a young mind, what examples can be given?

Prove: If $(g \circ f)$ is bijective, is $f$ bijective?

I need to prove or disprove for a discrete mathematics assignment the following statement: $(g \circ f)$ is bijective $\rightarrow$ $f$ is bijective, $f: X \rightarrow Y$ $\hspace{.5cm} g:Y\rightarrow Z$ All of the domains and codomains here are supposed to be the real numbers. I’m having a hard time understanding how to prove things about […]

Proving equivalence of Axiom of Choice

I am working on the following question concerning the axiom of choice and one of its many equivalences. Advice as to whether I am on the right track would be appreciated. As a preface, I have looked at most of the other ‘axiom of choice’ posts on math.stackexchange already so referencing me to them may […]

union of sets disjoint sets is equal to the union set difference

Let $A_1,A_2,\ldots$ be a sequence of subsets of $\mathcal{X}$. Let $B_1=A_1$ and define $$ B_n=A_n\setminus \bigcup_{i=1}^{n-1}A_i \text{ for } n=2,3,\ldots $$ Show that $$ \bigcup_{i=1}^n B_i = \bigcup_{i=1}^n A_i \text{ for all } n\in\mathbb{N} $$ I can see why this is, intuitively, but I can’t figure out how to construct a formal proof. I can […]

Infinite Set has greater or equal cardinality that of N

For any infinite set, we can find a 1-1 function (not necessary onto) from N (set of natural no.) to that set. The proof of this theorem I know using axiom of choice. Can we prove it without using Axiom of choice (not using axiom of replacement and GCH). If not this means this statement […]