91Ó°ÊÓ

Show that if \(A \subseteq B\) and there is an injection \(g: B \rightarrow A,\) then \(|A|=|B|\)

Prove that the set \(\mathbb{C}\) of complex numbers is uncountable.

Show that \(\left|\mathbb{R}^{2}\right|=|\mathbb{R}| .\) Suggestion: Begin by showing \(|(0,1) \times(0,1)|=|(0,1)|\).

Let \(\mathscr{F}\) be the set of all functions \(\mathbb{N} \rightarrow\\{0,1\\} .\) Show that \(|\mathbb{R}|=|\mathscr{F}| .\)

Show that the two given sets have equal cardinality by describing a bijection from one to the other. Describe your bijection with a formula (not as a table). The set of even integers and the set of odd integers

Prove or disprove: If \(A \subseteq B \subseteq C\) and \(A\) and \(C\) are countably infinite, then \(B\) is countably infinite.

Prove or disprove: There exists a countably infinite subset of the set of irrational numbers.

Prove or disprove: The set \(\mathbb{Q}^{100}\) is countably infinite.

Prove or disprove: If there is an injection \(f: A \rightarrow B\) and a surjection \(g: A \rightarrow B\), then there is a bijection \(h: A \rightarrow B\).

Prove or disprove: The set \(\left\\{\left(a_{1}, a_{2}, a_{3}, \ldots\right): a_{i} \in \mathbb{Z}\right\\}\) of infinite sequences of integers is countably infinite.