query: Let A be a countable set, and let $f : A \to B$ be a function. Prove that $f(A)$ is at most countable.
positive_passages.docid: 94365
positive_passages.text: How to prove that if $D$ is countable, then $f(D)$ is either finite or countable?
negative_passages.docid: 37398
negative_passages.text: Proof that$ (a+A)\cap A=\varnothing$ if $A$ is countable
negative_passages.docid: 93419
negative_passages.text: Let $\mathcal A$ be a collection of pairwise disjoint subsets of a $\sigma$-finite measure space,Show that $\mathcal A$ is at most countable.
negative_passages.docid: 42095
negative_passages.text: Existence in ZF of a set with countable power set
negative_passages.docid: 125043
negative_passages.text: Show that F(A−B) is not necessarily a subset of F(A)−F(B).
negative_passages.docid: 70143
negative_passages.text: Mutually exclusive countable subsets of a countable set
negative_passages.docid: 28391
negative_passages.text: Prove the compactness of a set B
negative_passages.docid: 18223
negative_passages.text: Prove that the union of two disjoint countable sets is countable
negative_passages.docid: 48765
negative_passages.text: Let $S$ and $T$ be finite non-empty sets such that $|S| = |T|$. Show that the function $f : S\to T$ is onto if and only if it is one-to-one.
negative_passages.docid: 147136
negative_passages.text: Subset of a countable set is itself countable
negative_passages.docid: 72694
negative_passages.text: Proving that $f(x) = g(x)$ for some $x \in [a,b]$ if $f,g$ continuous, $f(a)  g(b)$
negative_passages.docid: 79490
negative_passages.text: Proving the set of points at which a function diverges to $\infty$ is countable
negative_passages.docid: 40937
negative_passages.text: Is the set of partial order relations over $ \mathbb{N}$ countable?
negative_passages.docid: 40604
negative_passages.text: Is this proof correct for : Does $F(A)\cap F(B)\subseteq F(A\cap B) $ for all functions $F$?
negative_passages.docid: 110635
negative_passages.text: Proving $\mathbb{N}^k$ is countable
negative_passages.docid: 85612
negative_passages.text: Let functions $ f: A \to B$ and $g: B \to A$, suppose that $ g \circ f = i_A$. Prove that if $f$ is onto, then $g$ is one-to-one
negative_passages.docid: 91237
negative_passages.text: Every infinite set contains a countable subset