Start Learning Sets 2 | Predicates, Equality and Subsets [dark version]

Set-building in mathematics hinges on turning “open” statements with placeholders into precise logical statements, then collecting exactly the objects that make them true. Predicates—expressions with undetermined variables—become the engine for constructing new sets from old ones. For example, “n is an even number” isn’t automatically true or false until n is replaced by a specific number; once n = 2, the statement becomes true, while n = 1 makes it false. The same idea applies to mathematical forms like “y + 8 = 9”: it only gains a truth value after choosing a particular y.

With that logic in place, sets are defined using curly braces plus a predicate. The notation describes “the set of all X in some larger set that satisfy a condition.” So the set of natural numbers X such that X is even is precisely the collection of those elements that make the predicate evaluate to true. Another example uses membership: “the set of all Y in Z such that Y is an element of the natural numbers” filters integers down to those that belong to N. The method generalizes beyond math: the set of planets P such that P has at least one confirmed moon is formed by checking each planet against the predicate and keeping only those that pass.

Once predicates are in hand, quantifiers supply the missing structure for statements that talk about many objects at once. The universal quantifier “for all” (∀) means every object considered must satisfy the predicate; the existential quantifier “exists” (∃) means at least one object satisfies it. Combining a quantifier with a predicate yields a complete logical statement with a definite truth value. For instance, “for all X, X is a planet” is false because not every object in the chosen domain is a planet, while “there exists X such that X is a planet” is true when the domain includes at least one planet.

These tools then define fundamental set relations. Equality of sets A and B means they contain exactly the same elements: for every X, X is in A if and only if X is in B. The transcript illustrates this with a concrete example where two sets listing the same numbers in different orders are shown equal by checking membership for each possible object. Repetition of elements doesn’t change equality either, since sets are determined by membership, not listing order or duplicates.

Subset is defined similarly using a conditional: A is a subset of B if every element of A is also an element of B. Formally, for all X, if X is in A then X is in B. The same mirrored notation can be used depending on preference, but the meaning stays the same. Together, predicates, quantifiers, and these definitions provide the logical foundation for building and comparing sets—setting up more set operations for the next lesson.