Infinity And Complements

Probability Level 2

Suppose $X \subseteq \mathbb{N}$

Let $A$ be the statement that $X$ contains finitely many elements.

Let $B$ be the statement that the complement of $X$ , say $X'= \mathbb{N} \setminus X$ contains infinitely many elements.

Which of the following implications is/are true?

(1) : $A \implies B$ .
(2) : $B \implies A$ .

Notation : $\mathbb N$ denotes the set of natural numbers .

(1) only (2) only (1) and (2) Neither (1) nor (2)

1 solution

Agnishom Chattopadhyay
Jul 17, 2016

If $A$ , is true $B$ must be.

Proof: Assume not. So, this means that there are $m,n \in \mathbb{N}$ such that they are respectively the cardinalities of $X$ and its complement. This means that the cardinality of $\mathbb{N}$ is $| m + n |$ . But this cannot be true, since we know that the cardinality of $\mathbb{N}$ is greater than the cardinality of any $n \in \mathbb{N}$

$B$ does not imply $A$ .

Counterexample: Let X be the set of all odd integers. Then, both X and its complements are infinite.

Umm I had a query : what could be the complement of say a set X which has infinitely many elements in N ? Is it a null set ?

Arnav Das - 4 years, 11 months ago

Not necessarily.

Let X be the set of all odd integers. Then, both X and its complement [the set of all even integers] are infinite.

Agnishom Chattopadhyay - 4 years, 11 months ago

Small typo in the solution. Second line of proof : m should be $\mathbb{N}$

A Former Brilliant Member - 4 years, 10 months ago

Thanks. Fixed.

Agnishom Chattopadhyay - 4 years, 10 months ago

Infinity And Complements

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