List of set identities and relations

It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.

Unions, intersection, and symmetric difference are commutative operations:[3]

must hold by definition of equality), and so in this sense, set subtraction is as diametrically opposite to being commutative as is possible for a binary operation.

ordered by inclusion, is a bounded lattice, and hence together with the distributive and complement laws above, show that it is a Boolean algebra.

There is no universal agreement on the order of precedence of the basic set operators.

where the only difference between the left and right hand side set equalities is that the locations of

Union does not distribute over symmetric difference because only the following is guaranteed in general:

The great variety and (relative) complexity of formulas involving set subtraction (compared to those without it) is in part due to the fact that unlike

where the above two sets that are the subjects of De Morgan's laws always satisfy

(Recall that symmetric difference is associative so parentheses are not needed for the set

The binary Cartesian product ⨯ distributes over unions, intersections, set subtraction, and symmetric difference:

and: A consequence of this is the following assumption/definition: Some authors adopt the so called nullary intersection convention, which is the convention that an empty intersection of sets is equal to some canonical set.

However, the nullary intersection convention is not as commonly accepted as the nullary union convention and this article will not adopt it (this is due to the fact that unlike the empty union, the value of the empty intersection depends on

so if there are multiple sets under consideration, which is commonly the case, then the value of the empty intersection risks becoming ambiguous).

are labelled; and analogously, the left hand side depends on how for each fixed

Equality in Inclusion 1 ∪∩ is a subset of ∩∪ can hold under certain circumstances, such as in 7e, which is the special case where

For a correct formula that extends the distributive laws, an approach other than just switching

However, sometimes these products are somehow identified as the same set through some bijection or one of these products is identified as a subset of the other via some injective map, in which case (by abuse of notation) this intersection may be equal to some other (possibly non-empty) set.

denote the Cartesian product, which (as mentioned above) can be interpreted as the set of all functions

In words, preimages distribute over unions, intersections, set subtraction, and symmetric difference.

In words, images distribute over unions but not necessarily over intersections, set subtraction, or symmetric difference.

The example above is now generalized to show that these four set equalities can fail for any constant function whose domain contains at least two (distinct) points.

Mnemonic: In fact, for each of the above four set formulas for which equality is not guaranteed, the direction of the containment (that is, whether to use

(the resulting statement is always guaranteed to be true) because this is the choice that will make

Alternatively, the correct direction of containment can also be deduced by consideration of any constant

Furthermore, this mnemonic can also be used to correctly deduce whether or not a set operation always distribute over images or preimages; for example, to determine whether or not

) is always the same as the answer for this choice of (constant) function and disjoint non-empty sets.

Indeed, comparison of that example with such a proof suggests that the example is representative of the fundamental reason why one of these four equalities in statements (b) - (e) might not hold (that is, representative of "what goes wrong" when a set equality does not hold).

The following table lists some well-known categories of families of sets having applications in general topology and measure theory.

[13] The article on this topic lists set identities and other relationships these three operations.