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When and how a set can be defined as an universal set

A set which has all the elements in the universe of discourse is called a universal set.

More formally, a universal set, denoted by U, is a set that satisfies the following: ∀x x ∈U.

Three subset relationships involving empty set and universal set are listed below as theorems without proof.

Note that the set A in the next four theorems are arbitrary. So A can be an empty set or universal set.

Theorem 1: For an arbitrary set A A ⊆U.

Theorem 2: For an arbitrary set A ∅ ⊆A.

Theorem 3: For an arbitrary set A A ⊆A.