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GREYUIT [131]
4 years ago
10

Show by using "proof by contradiction" that the set of binary sequences {0,1}N is uncountable.

Mathematics
1 answer:
liraira [26]4 years ago
6 0

Answer:

You can prove this important result as follows:

Step-by-step explanation:

Let A be the set of all binary sequences, that is to say, \{0,1\}^{\mathbb{N}}. Suppose that A is a countable set. Then the elements of  A can be ordered as a sequence \{s_{1},s_{2}, s_{3},...\}, where each s_{i} is a binary sequence. The k\text{-th} digit of each sequence is expressed by s_{n}(k). Define the sequence s as follows:

s(k)=\begin{cases}1&\text{if}\,s_{k}(k)=0\\ 0 &\text{if}\,s_{k}(k)=1\end{cases}

Note that s differ from each s_{k} in at least one digit. Then s\neq s_n for all n\geq 1, then s\notin A. This contradicts the fact that A is the set of all binary sequences. Then A must be a uncountable set.

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