Sets, induction and elementary functions

Given $A=\{0,1,2,3\}$ and $B=\{2,3,5\}$, determine $A\cup B$, $A\cap B$, $A\setminus B$ and $B\setminus A$.

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Given $A=(-3,2]$ and $B=[0,4)$, determine $A\cap B$, $A\cup B$, $A\setminus B$ and the complement $B^c$ with respect to $\mathbb{R}$.

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Given $A=\{x\in \mathbb{R}:|x|<2\}$ and $B=\{x\in \mathbb{R}:x^2\ge 1\}$, determine $A\cap B$ and $A\setminus B$.

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Given $A=\{1,2\}$ and $B=\{0,1\}$, write $A\times B$ and $B\times A$ by listing their elements, and determine whether the pair $(0,1)$ belongs to each of the two sets.

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Determine the set of upper bounds, the supremum, the infimum, the maximum and the minimum of

$$A=[1,4)\cup \{5\}.$$

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Determine the supremum, infimum, maximum and minimum of

$$A=\left\{3-\frac{1}{n}:n\in \mathbb{N},n\ge 1\right\}.$$

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Determine the supremum, infimum, maximum and minimum of

$$A=\left\{(-1{)}^n\frac{n}{n+1}:n\in \mathbb{N},n\ge 1\right\}.$$

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Prove with the characterization that

$$\sup \left\{\frac{n}{n+2}:n\in \mathbb{N},n\ge 1\right\}=1$$

and determine whether the set has a maximum.

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Prove with the characterization that

$$\inf \left\{\frac{3n+1}{n}:n\in \mathbb{N},n\ge 1\right\}=3$$

and determine whether the set has a minimum.

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Prove that $\sqrt{5}$ is irrational.

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Prove by induction that, for every $n\ge 1$,

$$\sum _{k=1}^n(2k-1)=n^2.$$

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Prove by induction that, for every $n\ge 1$,

$$\sum _{k=1}^n{k}^3=\frac{n^2(n+1{)}^2}{4}.$$

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Prove by induction that, for every $n\in \mathbb{N}$,

$$\sum _{k=0}^n{2}^k=2^{n+1}-1.$$

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Prove by induction that $2^n\ge n+1$ for every $n\in \mathbb{N}$.

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Prove by induction that $n!\ge 2^{n-1}$ for every $n\ge 1$.

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Prove by induction that $3^n>2^n+n$ for every $n\ge 2$.

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Prove by induction that $5^n-1$ is divisible by $4$ for every $n\in \mathbb{N}$.

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Prove by induction that $n^3+5n$ is divisible by $6$ for every $n\in \mathbb{N}$.

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Determine the coefficient of $x^3$ in the expansion of $(2x-1{)}^5$.

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Determine the natural number $n\ge 2$ such that

$$\left(\genfrac{}{}{0ex}{}{n}{2}\right)=36.$$

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Determine the natural domain of

$$f(x)=\sqrt{x^2-9}+\frac{1}{x-5}.$$

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Determine the natural domain of

$$f(x)=\ln \frac{x+1}{3-x}.$$

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Determine the natural domain of

$$f(x)=\frac{\sqrt{2-x}}{\ln (x+3)}.$$

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Determine the natural domain of

$$f(x)=\arcsin \frac{x-1}{2}.$$

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Verify that the function

$$f(x)=\frac{x+3}{x-2},$$

defined on $\mathbb{R}\setminus \{2\}$, is injective; determine its range and its inverse.

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Verify that the function

$$f(x)=e^{2x}+1$$

is strictly increasing on $\mathbb{R}$; determine its range and its inverse.

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Determine whether the following functions are even, odd, or neither even nor odd:

$$f(x)=x|x|,g(x)=\cos x+x^2,h(x)=x^3+1.$$

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Sketch the graph of

$$f(x)=3-(x+1{)}^2$$

starting from that of $y=x^2$, indicating the transformations used, the vertex and the range.

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Solve

$${\left(\frac{1}{3}\right)}^{2x-1}<9.$$

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Solve

$$9^x-4\cdot 3^x+3>0.$$

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Solve

$${\log }_{\frac{1}{2}}(x-2)>-3.$$

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Solve

$${\log }_{2}x+{\log }_2(x-1)<1.$$

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Solve

$$\sqrt{2}\sin x+1=0.$$

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Determine the range of

$$f(x)=\frac{3}{x^2+2}.$$

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