Mathematics:Teaching, Learning and Exploring.
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(Part C Question: NET June-2015) Which of the following sets of functions are uncountable?
A. \(\{f: f:\mathbb{N} \to \{1,2\}\}\)
B. \(\{f: f:\{1,2\} \to \mathbb{N}\}\)
C. \(\{f: f:\{1,2\} \to \mathbb{N}, f(1) \leq f(2)\}\)
D. \(\{f: f:\mathbb{N} \to \{1,2\}, f(1) \leq f(2)\}\)
\(\textbf{Options A and D}\)
Options A and D: The cardinality of \(\{f: f:\mathbb{N} \to \{1,2\}\}\) and \(\{f: f:\mathbb{N} \to \{1,2\}, f(1) \leq f(2)\}\)
is \( 2^\mathbb{N}= c\) and so are uncountable.
Options B and C: The cardinality of \(\{f: f:\{1,2\} \to \mathbb{N}, f(1) \leq f(2)\}\) and \(\{f: f:\{1,2\} \to \mathbb{N}, f(1) \leq f(2)\}\) is
\(\mathbb{N} \times \mathbb{N} \) and so are countable.
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