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Tantely R.
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Set Theory
TutorMe
Question:

What is the cardinality of the set of functions from a given set $S$ of cardinality $n$ to the set $\{0, 1\}$?

Tantely R.
Answer:

Let $f$ be a function from $S$ to $\{0, 1\}$. We know that $f$ is completely determined by the images of each element of $S$. And for each element $x$ of $S$, there are 2 choices for $f(x)$, namely 0 and 1. Therefore there are $2^{n}$ functions.

Calculus
TutorMe
Question:

Find a function such that the slope of the tangent of its point at any given point in the domain is equal to the value of the function at that point.

Tantely R.
Answer:

We are to solve the differential equation $( \frac{dy}{dt} = y$), which family of solutions is given by $( \{e^t + C \colon C\in \mathbb{R}\}.$)

Algebra
TutorMe
Question:

Solve the equation $( x^{4}+ y^{4} = 1)$.

Tantely R.
Answer:

By the Fermat-Wiles theorem, the solutions of this equation are the trivial ones: $( (0, 1), (0, -1), (1, 0), (-1, 0))$

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