My textbook asks me to decide whether or not this expression is true: Given the function $f: X \to Y$ with $B_1 \subseteq Y $. $ f^{-1}(Y $ \ $ B_1) = X $ \ $f^{-1}(B_1) $ I was confused because there is no assumption that the function $f$ is onto. Hence there might be […]

Could you please offer a straightforward idea to demonstrate that the set of monotonically increasing sequences of natural numbers is indeed uncountable? I was asked to show the cardinality of this set but the best I can do right now is the following Monotonically increasing sequences are equivalent to infinite subsets (trivial). Number of finite […]

I am confused by the proof a proposition: $F$ is a free abelian group on a set $X$ and $H$ is a subgroup of $F$, then $H$ is free abelian on a set $Y$, where $|Y| \leq |X|.$ The proof is: Let $X$ be well-ordered in some fashion, say as $\{ x_{\alpha} | \alpha < […]

I found the following definition for lexicographical ordering on Wikipedia (and similar definitions in other places): Given two partially ordered sets $A$ and $B$, the lexicographical order on the Cartesian product $A \times B$ is defined as $(a,b) \le (a’,b’)$ if and only if $a < a’$ or ($a = a’$ and $b \le b’$). […]

A open set is a set that can be written as a union of open intervals. If $f$ is a real valued continuous function on $\mathbb{R}$ that maps every open set to an open set, then prove that $f$ is a monotone function.

The standard presentation of Cantor’s Diagonal argument on the uncountability of (0,1) starts with assuming the contrary through “reduction ad absurdum”. The intuitionist schools of mathematical regards “Tertium Non Datur” (bijection from N to R either exists or does not exist) untenable for infinite classes. However, the argument can be modified by applying the diagonal […]

How to prove that $$\forall Y \subset X \qquad \forall U \subset X \qquad(Y \setminus U = Y \cap (X \setminus U))$$ Intuitively it has something to do with the way $2^X$ models Boolean algebra, but I can’t make a precise argument. Any hint?

Let $R$ and $S$ be relations on a set $A$. Assume $A$ has at least three elements. These are my best guesses at these two proofs. The first one I don’t feel confident about at all, as it seems I’m making too many assumptions. If $R$ and $S$ are transitive, then $R \cap S$ is […]

I thought the set of natural number functions would be of the same cardinality as the countably infinite product of $\mathbb{N}$, which is countable. Each natural number function can be identified with an infinite-tuple of $\mathbb{N}$ by letting the $i$th entry be the image of the number $i$ under the function.

Enderton defines the rank of a set $A$ to be the least ordinal $\alpha$ such that $A \subseteq V_{\alpha}$ (equivalently, $A \in V_{\alpha^+}$). He the derives the following identity: $rank(A) = \bigcup \{ (rank(x))^+ : x \in A \}$ for all sets $A$. In Exercise 30 of Chapter 7, the reader is asked to prove […]

Intereting Posts

Cover $\{1,2,…,100\}$ with minimum number of geometric progressions?
Prove $2^{1/3}$ is irrational.
Calculating probabilities over different time intervals
Circle Rolling on Ellipse
A non-UFD where we have different lengths of irreducible factorizations?
Interesting log sine integrals $\int_0^{\pi/3} \log^2 \left(2\sin \frac{x}{2} \right)dx= \frac{7\pi^3}{108}$
What (and how many) pieces does the Banach-Tarski Paradox break a sphere into?
Group $G$ of order $p^2$: $\;G\cong \mathbb Z_{p^2}$ or $G\cong \mathbb Z_p \times \mathbb Z_p$
A unsolved puzzle from Number Theory/ Functional inequalities
Vakil's definition of smoothness — what happens at non-closed points?
Dominoes and induction, or how does induction work?
Show that $G$ is cyclic
Existence of isomorphism between tensor products.
A tough series related with a hypergeometric function with quarter integer parameters
Why is $e^{x}$ not uniformly continuous on $\mathbb{R}$?