2

Let $A$ be the set of all even integers. Let $B$ be the set of all odd integers.

How do we prove that $|A|=|B|$?

I understand that I need to establish a bijection, but how do I go about doing that?

Any advice would be really appreciated.

Community
  • 1
1011011010010100011
  • 709

3 Answers3

2

Take the map defined by $f(n)=n+1$ it defines a bijection between the even integers and odd integers.

Tsemo Aristide
  • 87,475
2

$$x\in A\leftrightarrow x+1\in B$$ $$\dots$$ $$-4\leftrightarrow-3$$ $$-2\leftrightarrow-1$$ $$0\leftrightarrow1$$ $$2\leftrightarrow3$$ $$4\leftrightarrow5$$ $$\dots$$

Parcly Taxel
  • 103,344
  • What exactly do you mean by this? Can you add some words to explain this. I am quite lost. – 1011011010010100011 Jul 19 '17 at 16:45
  • We associate with any number $x$ in set A (the evens) the number $x+1$ in set $B$ (the odds). End. – Parcly Taxel Jul 19 '17 at 16:46
  • I understand that, but how do I formally explain your original answer? I can't just show a few and write "..." and that be a formal proof. And in order to show bijection (in formally written proofs), I need to show surjection and injection... – 1011011010010100011 Jul 19 '17 at 16:48
  • @sgerbhctim There exists an inverse function $x\in\B\leftrightarrow x-1\in\A$, so it is already a bijection. – Parcly Taxel Jul 19 '17 at 16:50
1

Connect the nth odd number i.e 2n-1 with the nth even number i.e 2n in your bijection.

Haran
  • 9,717
  • 1
  • 13
  • 47
  • Besides, I think that if two sets are countably infinite, both of them have equal cardinality – Haran Jul 19 '17 at 14:33