Set Operations – Unions, Intersections, Complements, and More


A Set

A set is a group of elements. 

For example, set “A” could be a group of four elements:

A = {3, 9, 12, 15}

And set “B” could also be a group of four elements:

B = {3, 4, 12, 19}

Union

The union of two sets, denoted as “∪”, is simply the list of elements that are in set A or set B. For example, the union of set A and B would be written as:

A∪B = {3, 4, 9, 12, 15, 19}

Intersection

The intersection of two sets, denoted as “∩”, is the list of elements that are in both set A and B. The intersection of set A and set B would only include the two elements that are in both sets:

A = {3, 9, 12, 15}

B = {3, 4, 12, 19}

A∩B = {3, 12}

Relative Complement

The relative complement of two sets is simply the difference between two sets. 

For example, the “relative complement of B in A” gives us all of the elements in A that are not in B. We denote this as A \ B.

To find A \ B, we start with all of the elements in A and simply remove any elements from A that are also in B:

A = {3, 9, 12, 15}

B = {3, 4, 12, 19}

A \ B = {9, 15}

We could also find the “relative complement of A in B”, which gives us all of the elements in B that are not in A. We denote this as B \ A:

B = {3, 4, 12, 19}

A = {3, 9, 12, 15}

B \ A = {4, 19}

Universal Set

A universal set is a set that contains all of the possible elements of something we care about. For example, suppose we have set “U”, which is our universal set:

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Suppose we also have set “A”:

A = {1, 2, 3}

Absolute Complement

The absolute complement of A, denoted as A’, is all of the elements in the universal set that are not in A:

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

A = {1, 2, 3}

A’ = {4, 5, 6, 7, 8, 9, 10}

Subsets, Null Sets, and Equal Sets

If every element in set A is also in set B, then set A is a subset of set B. Suppose A = {1, 2, 3} and B = {1, 2, 3, 4, 5}. In this case, set A is a subset of set B because every element in A is also in B. 

A set that contains no elements is called a null set or an empty set, and is usually written as {}. Suppose A = {1, 2, 3} and B = {4, 5, 6}. Recall that the intersection of these two sets is the set of numbers that are in both A and B. In this case, they share no common elements so the intersection would be {}, an empty set.

Two sets are equal if they share all of the same elements. For example, if set A = {14, 8, 2} and set B = {14, 8, 2}, then sets A and B are equal. 

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