Two experiments are described that investigate the ability to infer the number of items in oneto- one corresponding sets for two age groups. We assess the influence of set-size, the visibility of sets, and the way in which set equivalence is derived - pairing versus sharing - using a repeated-measures design. Three-year-olds are largely restricted to inferring number after separating out conceptually paired items. In contrast, four-year-olds are able to make appropriate inferences about shared items, but they typically prefer to count those items if they are visible. Moreover, the size of corresponding sets affects children’s propensity to count rather than infer. Children count more often on larger sets. The ability to infer number using cardinal extension is associated most strongly with sharing proficiency, although counting skills also play an important part. We discuss how the data reveal an emerging understanding of the relationship between one-to-one correspondence and cardinality.