Exercise · ZW-EQUIPOTENCE-010 · ★★★ Advanced
Let \(f\colon A\to B\) be a bijection, with \(A\) finite. Why is \(B\) also finite?
Choose one option.
A Because \(B\) has exactly the same elements as \(A\) B Because \(f\) maps \(A\) into \(B\), so \(B\) is larger C Because the image of \(f\) is always contained in \(A\) D Because the inverse \(f^{-1}\) composed with a bijection \(A\to\{1,\dots,n\}\) gives a bijection \(B\to\{1,\dots,n\}\) Show answer D. Because the inverse \(f^{-1}\) composed with a bijection \(A\to\{1,\dots,n\}\) gives a bijection \(B\to\{1,\dots,n\}\)← Back to the question
Explanation If \(A\) has \(n\) elements , there is a bijection \(\varphi\colon A\to\{1,\dots,n\}\). Then \(\varphi\circ f^{-1}\) is a bijection \(B\to\{1,\dots,n\}\), so \(B\) is finite.
Two sets \(A\) and \(B\) are equipotent if there is a bijection \(f\colon A\to B\). It is written \(A\sim B\).
\[A\sim B \iff \exists\, f\colon A\to B\ \text{biyectiva}\]
More practice problems: equipotence ZW-EQUIPOTENCE-001 What is needed for two sets A and B to be equipotent?ZW-EQUIPOTENCE-002 Is \a,b,c\ equipotent to \1,2,3\?ZW-EQUIPOTENCE-003 Which of these properties does equipotence satisfy?ZW-EQUIPOTENCE-004 Let ℕ=\0,1,2,dots\. Is ℕ equipotent to \0,2,4,dots\?ZW-EQUIPOTENCE-005 Which statement about ℕ and ℤ is correct?ZW-EQUIPOTENCE-006 How is it written that the sets A and B are equipotent?ZW-EQUIPOTENCE-007 Is \1,2\ equipotent to \1,2,3\?ZW-EQUIPOTENCE-008 What do you get when composing two bijections fcolon A→ B and gcolon B→ C?