Isabella Moreno (@moreno223)

What is the difference between Eulerian and Hamiltonian paths in graph theory?

I am studying graph theory and I keep confusing Eulerian paths with Hamiltonian paths. Can someone clarify the difference?

Eulerian path: A path that uses every edge exactly once.
Hamiltonian path: A path that visits every vertex exactly once.

1691Discrete Mathematics Combinatorics

How to solve a first-order linear ODE using an integrating factor?

I'm studying differential equations and I keep making mistakes when using the integrating factor method.

For a first-order linear ODE of the form:

1342Linear Algebra Combinatorics

How to prove a function is injective, surjective, or bijective?

I'm studying discrete mathematics and I need to understand the three types of functions. **Definitions:** - **Injective (one-to-one):** $f(x_1) = f(x_2) \implies x_1 = x_2$ - **Surjective (onto):** For every $y$ in the codomain, there exists $x$ such that $f(x) = y$ - **Bijective:** Both injective and surjective

1294Mathematical Logic Abstract Algebra

How to integrate ∫ e^x sin(x) dx using integration by parts?

I am studying AP Calculus BC and I keep going in circles when trying to evaluate this integral: $$\int e^x \sin(x) \, dx$$

1346Number Theory Combinatorics

Why does the order of operations matter in math?

PEMDAS or BODMAS is drilled into every student, but why does the order matter at all? If math describes the real world, shouldn't the answer be the same regardless of the order you compute? Who decided that multiplication comes before addition?

1134Algebra Math Misconceptions

Why is the golden ratio considered beautiful by many?

The golden ratio phi appears in art, architecture, and even the proportions of the human face. People claim it's naturally pleasing to the eye. Is this backed by real science or is it mostly a myth that keeps getting repeated without evidence?

1115Geometry Math Misconceptions

How to solve recurrence relations using the characteristic equation method?

I am studying discrete mathematics and I need to solve linear recurrence relations. For example: $$a_n = 5a_{n-1} - 6a_{n-2}, \quad a_0 = 1, a_1 = 2$$

1262Discrete Mathematics Combinatorics

How to prove a Cauchy sequence converges in ℝ?

I am taking an introductory Real Analysis course and we are studying the Cauchy criterion for convergence. I understand the definition: A sequence $(a_n)$ is Cauchy if for every $\varepsilon > 0$, there exists $N \in \mathbb{N}$ such that for all $m, n \geq N$, $|a_n - a_m| < \varepsilon$.

1530Linear Algebra IIT-JEE Mathematics

How to use the inclusion-exclusion principle for counting problems?

I am studying combinatorics and I understand the basic inclusion-exclusion principle for two sets: $$|A \cup B| = |A| + |B| - |A \cap B|$$

1369Discrete Mathematics Combinatorics

How to find the number of solutions to a linear Diophantine equation?

I am studying number theory and I need to understand Diophantine equations — equations where only integer solutions are allowed. Consider: $3x + 5y = 17$

1268Number Theory Discrete Mathematics

How to prove that the derivative of sin x is cos x using the limit definition?

I'm in a calculus class and we proved that $\frac{d}{dx} \sin x = \cos x$ using the limit definition of the derivative: $$\frac{d}{dx} \sin x = \lim_{h \to 0} \frac{\sin(x+h) - \sin x}{h}$$

1366Mathematical Logic A-Level Mathematics

What is the difference between eigenvalue and eigenvector in intuitive terms?

I am studying linear algebra and I understand the formal definition: For a square matrix $A$, if $A\mathbf{v} = \lambda\mathbf{v}$ for some scalar $\lambda$ and non-zero vector $\mathbf{v}$, then $\lambda$ is an eigenvalue and $\mathbf{v}$ is the corresponding eigenvector.

1363Algebra Abstract Algebra

What is the difference between Eulerian and Hamiltonian paths in graph theory?

I am studying graph theory and I keep confusing Eulerian paths with Hamiltonian paths. Can someone clarify the difference?

Eulerian path: A path that uses every edge exactly once.
Hamiltonian path: A path that visits every vertex exactly once.

1691Discrete Mathematics Combinatorics

How to solve a first-order linear ODE using an integrating factor?

I'm studying differential equations and I keep making mistakes when using the integrating factor method.

For a first-order linear ODE of the form:

1342Linear Algebra Combinatorics