# Tagged Questions

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### Prove the reflexivity of $\subseteq$.

My professor gave me a list of exercises, I've been able to figure out what mechanism I should exploit to prove them, but I'd like to know if it's good. Until now we've been taught a little logic and ...
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### Composition of injections (proof)

I'm trying to prove that composition of injections is an injection. I want to know if this is a good proof: Composition of injections is an injection. Let $f:S_1\rightarrow S_2$ and ...
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### In a class of 65, there are twice as many maths students as biology students.

I have a task: In a class of 65, there are twice as many maths students as biology students. If 12 biology students do not take maths and 15 students take neither of these subjects, how many students ...
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### Subsets and Cardinality

I'm confused on if I should count a subset as one element or if I should count all the elements of that subset when computing cardinality. Example: Given the set $A = \{1,2,3,\{4,5,6\}\}$ does $A$ ...
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### Does $f^{-1}(Y)$ make sense if $Y$ is “bigger” than $X$

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 ...
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### How can I show that this has a reflexive closure?

Assume R is a relation on A. Show that R ∪ ΔA is the reflexive closure of R. What I thought is, that if R is an union with ΔA, it must mean that all a in A are (a,a). But I don't know if this is ...
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### Problem on number of elements in sets

This question in my text book chapter named "Permutation, Combination and Probability". But I am stuck with Permutation, Combination and probability. All things are seems same to me. As I am new in ...
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### Why is this relation irreflexive? And how can I prove it?

Why is the relation R on A irreflexive if and only if ΔA ∩ R = ∅? I always thought the empty set is reflexive (and transitive, symmetric because it is vacuously true.)
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### Zermelo-Fraenkel Set Theory Question regarding Ordered Pairs [closed]

Problem: Show that $\{x,\{y\}\}$ is not suitable as a definition of the ordered pair $(x,y)$, because it does not have the ordered pair property: For any sets $x,y,u,v$ if $( x,y ) = ( u,v )$, then ...
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### How many curly brackets of the empty powerset?

The question is: How many curly brackets are there in the following, if $\varnothing$ counts as $\{ \}$ (=1 curly brackets) $$\wp^5(\varnothing)$$ I calculated $p^2$, which was $4$ curly brackets, ...
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### For which sets, $X$ the relation is a partial function

Given $T=\left\{\ \left<A,B\right> \in (P(X))^2 | A\subseteq B \right\}$ For which sets, $X$, the relation $(P(X))^2-T \cap (P(X))^2-T^{-1}$ is a partial function? Here's my solution: ...
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### 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 ...
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### show that for an infinite cardinal $k$, $k + k = k$

Show that for an infinite cardinal $k$, $k + k = k$ So far I have that $k + k = 2k$ Is it possible to somehow show that $2k = k$? I've been trying to understand some cardinal arithmetic, and I ...
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### How to prove $A \cup \{a\} \approx B \cup \{ b \} \Rightarrow A \approx B$

How to prove this without recurring to cardinality? $A \cup \{a\} \approx B \cup \{ b \} \Rightarrow A \approx B$ Where by "$\approx$" I mean that there exists a bijective function between A and ...
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### Question about partitions in intervals of the real numbers.

I have to prove the following: Let $\mathcal{D}$ be a partition of $\mathbb{R}$ in intervals of any kind, except intervals containing a single element. Prove $\mathcal{D}$ is countable. ...
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### How do I write a rigorous proof of the following problem: $(A \triangle B) \cup C \neq (A \cup C) \triangle (B \cup C)$

Question: How do I write a rigorous proof of the following problem: Does the following equality hold true for any sets $A$, $B$ and $C$: $$(A \triangle B) \cup C = (A \cup C) \triangle (B \cup C)$$ ...
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### A problem concerning a partially ordered in $\omega$ and two chains.

Assume that $P=\left\langle\omega, \preceq\right\rangle$ is a partially ordered set such that for each $n \in \omega$ there are two chains $A_n$ an $B_n$ in $P$ such that $n \subset A_n \cup B_n$. ...
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### Prove there exists a countably infinite subset $A$ of $P(\mathbb{N})$ that satisfies given conditions [duplicate]

Prove that there exists a countably infinite set $A \subseteq$ $P(\mathbb{N})$ that satisfies all of the following conditions: $i)$ $X \cap Y = \emptyset$ for all $X, Y \in A$ such that $X \neq Y$ ...
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### the set of all of infinite N subsets

I was thinking about this one for a while and can't find a proper function. let $M=${$A\in P(N)$|A And A' are infinite} (for example: the set of all even numbers will be in M, the complement of A ...
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### If $f \colon A \to B$, $g :\colon B \to C$ and $g\circ f \colon A \to C$ are bijections. Prove that $f$ is 1-1, $g$ is onto.

From what I understand, one-to-oneness means every element in $A$ is mapped to a unique element in $B$. To be onto, means for every $y$ in $B$, there exist at least one $x$ in $A$ from which it can ...
### Prove that $A \cup B = A$ if and only if $B$ is a subset of $A$
If $A \cup B = A$ then $A$ is a subset of $A$ and $B$ is a subset of $A$. Thus $A \cup B = A$. If $B$ is a subset of $A$ then it follows that $A \cup B$ is a subset of $A$. My solution. It seems ...