Proof Tutorial Class, Week 24

Question One §

Question §

How does the graph of the function $f(x)$ change under the following transformations (where $a$ is a constant)? That is, how to obtain the graph of $F(x)$ from the graph of $f(x)$, where…

1. $F(x)=f(x)+a$;
2. $F(x)=f(x+a)$;
3. $F(x)=f(ax)$.

Solution §

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Question Two §

Question §

Assume the mapping $f:[0,1)\to [1,\infty )$, $f(x)=\frac{1}{\sqrt{1-x^2}}$, is a bijection. Apply transformations from the previous question to the mapping $f$ to produce a bijection $F:[2,4)\to [0,\infty )$.

Solution §

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Question Three §

Question §

Prove that the set $W$ of all finite sequences (‘words) composed of two letters $A,B$ is countable.

Hints: One method is to use a theorem in the lectures by which it is sufficient to produce an injective mapping to $\mathbb{N}$. Another method is to describe a systematic enumeration of all such words, that is, to show that the set $W$ can be represented as a sequence.

Solution §

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