Definition:Universal Affirmative/Set Theory

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The universal affirmative $\forall x: S \left({x}\right) \implies P \left({x}\right)$ can be expressed in set language as:

$\left\{{x: S \left({x}\right)}\right\} \subseteq \left\{{x: P \left({x}\right)}\right\}$

or, more compactly:

$S \subseteq P$

Also defined as

Some sources give this rule as:

$S \cap \complement \left({P}\right) = \varnothing$

where $\complement \left({P}\right)$ denotes the complement of $P$:

There are no objects which are $S$ which are not also $P$.

This is justified from Intersection with Complement is Empty iff Subset.

The advantage to this approach is that it allows the complete set of categorical statements to be be defined using a combination of set intersection and set complement operators.

Also see