Intereting Posts

If a plane is divided by $n$ lines, then it is possible to color the regions formed with only two colors.
Expected Value Function
Why is “the set of all sets” a paradox?
Geometric difference between two actions of $GL_n(\mathbb{C})$ on $G\times \mathfrak{g}^*$
Vector derivation of $x^Tx$
If $\mid a_{jj}\mid \gt \sum_{i \neq j} \mid a_{ij} \mid$ then vectors $a_1,\dots ,a_n \in \mathbb{R}^n$ are linearly indendent.
Isomorphism between quotient rings over finite fields
Evaluate definite integral $\int_{-1}^1 \exp(1/(x^2-1)) \, dx$
Why $2\frac{X_1}{X_1+X_2}-1$ and $X_1+X_2$ are independent if $X_1$ and $X_2$ are i.i.d. exponential?
$\lim_{p\rightarrow\infty}||x||_p = ||x||_\infty$ given $||x||_\infty = max(|x_1|,|x_2|)$
book suggestion on module theory
$R^n \cong R^m$ iff $n=m$
When is $5n^2+14n+1$ a perfect square?
Gaussian-Like integral
Mathematical induction proof that $\sum\limits_{k=2}^{n-1} {k \choose 2} = {n \choose 3} $

Let $L$ denote the set of all lines in the Euclidean plane $\Bbb R^2$. We say that two lines $L$ and $M$ are parallel if either $L = M$ or $L$ and $M$ have no points in common.

Using Euclid’s Parallel Postulate prove that parallelism is an equivalence relation on the set $L$.

I need some help determining if I have the right idea about this.

- Groups with only one element of order 2
- Is $\mathbb Q(\sqrt{2},\sqrt{3},\sqrt{5})=\mathbb Q(\sqrt{2}+\sqrt{3}+\sqrt{5})$.
- Generalization of irreducibility test .
- Does the equation $x^4+y^4+1 = z^2$ have a non-trivial solution?
- Infinite linear independent family in a finitely generated $A$-module
- What is $\gcd(0,a)$, where a is a positive integer?

$L=L$ which means that $L$ is parallel to itself. So this is reflexive.

Suppose we have $L || M$. Then Either $L=M$ or $L$ and $M$ share no points. So Then $M=L$ or $M$ and $L$ share no points. Then $M||L$. So this is symmetric.

Suppose we have $L || M$ and $M || N$. So then either $L=M$ or $L$ and $M$ share no points and either $M=N$ or $M$ and $N$ share no points. So we have that either $L=N$ or $L$ and $N$ share no points. So $L || N$. So then this is transitive.

So from all of this, parallelism is an equivalence relation. I’m not sure if this is correct or not. Any help would be appreciated.

- What is the difference between homomorphism and isomorphism?
- Why does the natural ring homomorphism induce a surjective group homomorphism of units?
- Find a polynomial of degree > 0 in $\mathbb Z_4$ that is a unit.
- Prove that any finite group $G$ of even order contains an element of order $2.$
- Showing that the only units in $\mathbb{Z}$ are $1,\, -1, \, i, \, -i$?
- What's the correct naming and notation those algebraic structures?
- Making sense out of “field”, “algebra”, “ring” and “semi-ring” in names of set systems
- Suppose $n$ is an even positive integer and $H$ is a subgroup of $\mathbb Z/n \mathbb Z$. Prove that either every element of $H$ is…
- What are a few examples of noncyclic finite groups?
- presentation of the direct sum of commutative rings / algebras

Your reasoning looks good. It might help to note explicitly that you are using symmetry of these two *other* relations, “$=$” and “share no points”, to conclude, for example:

$$L = M \implies M = L$$

and

$$\text{$L$ and $M$ share no points} \implies \text{$M$ and $L$ share no points}$$

ditto with reflexivity, transitivity.

In fact, these relations are both equivalence relations, and your proof is an expression of the fact that you can form a new equivalence relation from two old equivalence relations by joining them together with “or”.

But that’s a tangent. The primary benefit in being really pedantic and explicit about what facts you are using in these sorts of proofs, IMO, is the confidence boost.

- Pullback distributes over wedge product
- Existence of smooth function $f(x)$ satisfying partial summation
- Singularity at infinity of a function entire
- Find the sum of the series $\sum \frac{1}{n(n+1)(n+2)}$
- Numerical Analysis References
- Number of inversions
- Proof that the harmonic series diverges (without improper integrals)
- Solve the matrix equation $X = AX^T + B$ for $X$
- Evaluating $\lim_{n\rightarrow\infty}x_{n+1}-x_n$
- How to place objects equidistantly on an Archimedean spiral?
- Meaning of the identity $\det(A+B)+\text{tr}(AB) = \det(A)+\det(B) + \text{tr}(A)\text{tr}(B)$ (in dimension $2$)
- Ordinal Exponentiation and transfinite induction
- Can someone clearly explain about the lim sup and lim inf?
- If $xy+xz+yz=3$ so $\sum\limits_{cyc}\left(x^2y+x^2z+2\sqrt{xyz(x^3+3x)}\right)\geq2xyz\sum\limits_{cyc}(x^2+2)$
- $H_0(X) \cong \tilde{H}_0(X) \oplus \mathbb{Z}$ isomorphism in algebraic topology