Algebra of sets

In mathematics, especially in the area of set theory, the algebra of sets helps us understand how groups of items, called sets, can be combined and compared. This algebra looks at three main ways to bring sets together: union, where we combine items from both sets; intersection, where we find items common to both sets; and complementation, which looks at items not in a set.

These operations follow special rules, much like the rules we use in regular math when we add or multiply numbers. By studying these rules, mathematicians can solve problems and prove new ideas about sets.

When sets and their operations follow these rules, they form something called a Boolean algebra. In this special case, union works like joining sets together, intersection finds common parts, and complement shows what is missing. There is also a smallest set, empty set ⁠ ∅, and a largest set, called the universe set, which contains everything we are considering.

Fundamentals

A set is a group of different things, like numbers, symbols, points in space, lines, shapes, or even other groups. We usually use a big letter to name a set and show its items inside curly braces. For example, one set might be A = {1, 2, 3, 4} and another might be B = {a, b, c, x, y, z}.

The algebra of sets looks at how sets work together using two main ways to combine them: union and intersection. A union puts together all items from two or more sets into one big set. An intersection finds only the items that are in both sets. For example, if A = {1, 2, 3, 5} and B = {2, 4, 6}, then A together with B is {1, 2, 3, 4, 5, 6}, and A shared with B is just {2}.

These ways of combining sets have special rules, much like how numbers work in arithmetic. For example, the order doesn't matter when we combine sets, just like how adding 2 + 3 is the same as 3 + 2. Sets also follow other rules that help us understand how they fit together.

Main articles: Union (set theory) and Intersection (set theory)

Main articles: Empty set, Universe set, and Complement (set theory)

Principle of duality

See also: Duality (order theory)

In set algebra, there is a special rule called the principle of duality. This rule tells us that for any true fact about groups of things (sets), we can create another true fact by switching two main operations: combining groups (union) and finding common items (intersection). We also switch two special groups: the empty group (∅) and the whole group we are looking at (U). If we do these switches and also flip any ideas about one group being inside another, the new statement will also be true.

A statement is called self-dual if it stays the same even after we make these switches. This principle helps us understand how sets work in a balanced way.

Some additional laws for unions and intersections

For any groups or sets called A and B inside a bigger set called U, there are special rules that help us understand how these sets work together.

Some of these rules include:

• Combining a set with itself gives the same set.
• Finding parts of a set that are also in another set can be done in a special way using what’s called "set difference".

These rules make working with sets easier and more organized.

Some additional laws for complements

Let A and B be parts of a group called the universe U. There are special rules for finding the complement of sets, which is like finding what is missing from a set to make up the whole universe.

One important rule is de Morgan's laws. These tell us that the complement of two sets joined together is the same as finding the complements of each set separately and then joining them in a different way. Another rule is the double complement law, which says that if you take the complement of a set and then take the complement again, you get back the original set. There are also rules for the universe set and an empty set, showing that the complement of everything is nothing, and the complement of nothing is everything.

Algebra of inclusion

The algebra of sets includes the idea of one set being inside another set. This is called inclusion. When we say a set A is included in set B, we mean every item in A is also in B. We call A a subset of B and write it as A ⊆ B.

There are three main rules for inclusion:

1. Reflexivity: Every set is a subset of itself. So A ⊆ A.
2. Antisymmetry: If A is a subset of B and B is a subset of A, then A and B are exactly the same set.
3. Transitivity: If A is a subset of B and B is a subset of C, then A is also a subset of C.

These rules help us understand how sets relate to each other. For example, the smallest possible set is the empty set ∅, and the largest set is the whole universe of items we are looking at. Inclusion helps us see how smaller groups fit inside bigger groups.

Algebra of relative complements

The following proposition lists several identities concerning relative complements and set differences.

PROPOSITION 9: For any universe ⁠ U and subsets ⁠ A ⁠, ⁠ B ⁠ and ⁠ C ⁠ of ⁠ U ⁠, the following identities hold:

• ⁠ C ∖ ( A ∩ B ) = ( C ∖ A ) ∪ ( C ∖ B )
• ⁠ C ∖ ( A ∪ B ) = ( C ∖ A ) ∩ ( C ∖ B )
• ⁠ C ∖ ( B ∖ A ) = ( A ∩ C ) ∪ ( C ∖ B )
• ⁠ ( B ∖ A ) ∩ C = ( B ∩ C ) ∖ ( A ∩ C ) = ( B ∩ C ) ∖ A = B ∩ ( C ∖ A )
• ⁠ ( B ∖ A ) ∪ C = ( B ∪ C ) ∖ ( A ∖ C )
• ⁠ ( B ∖ A ) ∖ C = B ∖ ( A ∪ C )
• ⁠ A ∖ A = ∅
• ⁠ ∅ ∖ A = ∅
• ⁠ A ∖ ∅ = A
• ⁠ B ∖ A = A ∁ ∩ B
• ⁠ ( B ∖ A ) ∁ = A ∪ B ∁
• ⁠ U ∖ A = A ∁
• ⁠ A ∖ U = ∅