“It is said that a set can be defined connotatively or denotatively. Which of these terms applies to the definition by roster and which to the definition by a defining sentence?”- p. 137, Elements of Modern Math, K. O. May.
The author is asking me to match the words connotative and denotative to the roster notation $\{\,\}$ or to a a set designated by a defining sentence $\{ x \mid f(x) \}$. I say that the roster notation is connotative and a defining sentence of a set is denotative. The former lists the elements of a set, and generally what could belong in that set, while the latter defines a set, kind of, I suppose as a dictionary does.
Is this correct? Any elucidating thoughts are welcome.
Actually, it is the other way round.
The connotation (also known as intension) of a set is its description or defining properties, i.e., what is true about the elements of a set.
The denotation (also known as extension) of a set is its content, i.e., the list of its elements.
A set is totally defined by its denotation, and the same denotation (i.e., the same set) can have several connotations. For instance, the set $\{2, 3\}$ can be described as:
For an infinite set, it is not possible to explicitly give its denotation (since the list of its elements is infinite), the only way to identify it is connotative.