Number of maps (TIFR 2014 problem 30)

Number of maps (TIFR 2014 problem 30)

Question: How many maps \(\phi: \mathbb{N} \cup \{0\} \to \mathbb{N} \cup \{0\}\) are there satisfying \(\phi(ab)=\phi(a)+\phi(b)\) , for all \(a,b\in \mathbb{N} \cup \{0\}\) ? Discussion: Take \(n\in \mathbb{N} \cup \{0\} \). By the given equation \(\phi(n\times...
Maps from compact spaces (TIFR 2014 problem 29)

Maps from compact spaces (TIFR 2014 problem 29)

Question: Let \(f: X\to Y \) be a continuous map between metric spaces. Then \(f(X)\) is a complete subset of \(Y\) if A. X is compact B. Y is compact C. X is complete D. Y is complete Discussion: Let \(X\) be compact. Then \(f(X)\) is compact. (continuous image of...