In: Advanced Math
1. Use cardinality to show that between any two rational numbers there is an irrational number. Hint: Given rational numbers a < b, first show that [a,b] is uncountable. Now use a proof by contradiction.
2. Let X be any set. Show that X and P(X) do not have the same cardinality. Here P(X) denote the power set of X. Hint: Use a proof by contradiction. If a bijection:X→P(X)exists, use it to construct a set Y ∈P(X) for which Y is not in the range of f.
Question no. 1
Question no. 2
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