The term "universe of discourse" generally refers to the entire set of terms used in a specific discourse, i.e. the family of linguistic or semantic terms that are specific to any one area of interest. In semantics, discourses are linguistic units composed of several sentences - in other words, conversations, arguments or speeches. ... In general, semantics (from the Greek semantikos, or significant meaning, derived from sema, sign) is the study of meaning, in some sense of that term. ...
When the universe of discourse relates to a common experience, but this experience is of something imaginary, as when we discuss the world of Shakespeare's creation in the play of Hamlet, we find individual distinction existing so far as the work of imagination has carried it, while beyond that point there is vagueness and generality.
Our discourse seldom relates to this universe: we are either thinking of the physically possible, or of the historically existent, or of the world of some romance, or of some other limited universe.
"The universe of discourse is the aggregate of the individual objects which "exist," that is are independently side by side in the collection of experiences to which the deliverer and interpreter of a set of symbols have agreed to refer and to consider.
Here, the phrase "for all" implicitly requires a universe of discourse to specify which mathematical objects are "all" the possibilities for x.
If the universe of discourse consists of all functions from the real line R to itself, then the solutions for f are all functions whose only values are one and zero.
But if the universe of discourse consists of all continuous functions from R to itself, then the solutions for f are only the constant functions with value one or zero.
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