# 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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### Proof Verification for Discrete Math Class

Prove that $n^2$ is even iff $n$ is even. I proved it like this: Case I: $n$ is even 1) $n = 2a$ $(a\in Z)$ 2) $n^2 = 4a^2 = 2(2a^2)$ 3) $2a^2 = K$ $(K \in Z)$ 4) $n^2 = 2K$ Case II: $n$ is ...
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### Find the problem with this proof.

The following attempts to prove that if $n^2$ is even, then $n$ is even. Suppose $n^2$ is even. Then $n^2$ = 2$k$ for some integer $k$. Let $n$ = 2$m$ for some integer $m$. This shows that $n$ is ...
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I need to prove by contradiction that $\log_2(3)$ is irrational. I'm really unfamiliar with logs to be honest, it's been awhile since I've done them and I'm unsure of how to approach this. Any help ...
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### Proof by contradiction proving both numbers are not odd.

I have to do a proof by contradiction: Suppose $a,b,\in\mathbb{Z}$. If $4| (a^2 + b^2)$ then a and b are not both odd. So far I know that I need to prove that if $4|(a^2+b^2)$ then a and b are both ...
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### An undirected graph $G$ can be decomposed into simple edge-disjoint cycles if and only if all of its vertices have even degree.

Research effort: $\rightarrow)$ I think this is relatively easy. $\leftarrow)$ Let $G = (V,E)$, let $w$ be any vertex of $G$, given that all the vertex have even degree, I'm assured that I can ...
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### Proof regarding division with a remainder

Let $a\in\mathbb{Z},n\in\mathbb{N}$. If $a$ has a remainder $r$ when divided by $n$, then $a\equiv r\pmod n$ I've done some of these questions before with modulus and division, but I'm unsure of how ...
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### Proof regarding divisibility

if $n\in\mathbb{Z}$, then $4$ does not divide $(n^2 - 3)$ I'm not sure how to approach this question, I know how to do questions that involve proving that it does divide but I'm unsure of how to do ...
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### Modulus related proof help

I need to prove this via either direct proof, or contrapositive. Unsure of the best way to approach this. if $a \equiv b\mod n$ and $c \equiv d\mod n$, then $ac \equiv bd\mod n$ So far I have: ...
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### Proof about isometries, symmetry and reversing orientation.

Suppose $f:\mathbb{R} \rightarrow \mathbb{R}$ is an isometry of the reals. Prove that $f$ is a symmetry around a point if and only if $f$ reverses orientation of $\mathbb{R}$. The orientation of ...
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### If a connected graph has a unique spanning tree, then it is a tree.

Prove if a connected graph has a unique spanning tree, then it is a tree. Edit: This can be shown with proof by contradiction.
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### Is this proof by induction correct?

Prove by induction that for all $n\in\mathbb N$, $3\mid n^3+3n^2+2n$. $$P(1) = (1)^3+3(1)^2+2(1) = 6$$ Which is clearly divisble by $3$. Therefore, $P(1)$ is true. Assume $P(1),\ldots,P(n)$ and ...
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### Let $G$ be a graph with radius $r$ and let $x$ be a vertex in the center of $G$. If $d$($x,y$)=$r$, then $y$ is in the periphery of $G$

Prove, disprove, or give a counterexample: Let $G$ be a graph with radius $r$ and let $x$ be a vertex in the center of $G$. If $d$($x,y$)=$r$, then $y$ is in the periphery of $G$. Eccentricity - the ...
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### If a tree has order 2 or more, then the minimum cut set is 1. [closed]

Prove: If a tree has order 2 or more, then the minimum cut set is 1.
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### Suppose $f:X\to Y$ is onto and $A\subseteq Y$. Then $f(f^{-1}(A))=A$.

Prove, disprove, or give a counterexample: Suppose $f:X\to Y$ is onto and $A\subseteq Y$. Then $f(f^{-1}(A))=A$. Edit: Does this work? Suppose $f:X \to Y$ is onto and $A \subseteq Y$. We know ...
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### Proof that any self-complementary graph has to have $4k$ or $4k+1$ vertex, for some $k \in \mathbb N$

I've seen looking on previous questions that using this algorithm, I can construct self-complementary graphs. I got confused at this point, because I'm not sure if I should proof that there are no ...
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### Finding quantity of vertex of odd degree in a graph

Let $G=(V,X)$. If $G$ has n vertex, such exactly $n-1$ have odd degree, how many vertex of odd degree have $\overline {G}$. ($\overline {G}$ the complement of $G$.) So the first thing I notice is ...
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### Proof that in a graph of $2$ or more vertrex, there's at least $2$ of them that have the same degree

That's what I what to prove let $G$ be an undirected graph such it has $2$ or more vertex, there's at least $2$ vertex that have the same degree. I'm almost certain that, supposing that it's possible ...
First of all, here is the question I am trying to answer for context. I can see that the statement $\forall x \in \mathbb{Z} , \exists y \in \mathbb{Z}((x\leq y ) \wedge (x+y=0))$ negates to ...