A set is a set of different objects with a common property.
So about the amount of fruit in a basket or the amount of integers that are included in a list.

If the numbers \(0, 1, 4, 9\) are contained in a set \(A\), write \(A = {0,1,4,9}\).

If a number is contained in a set, for example the \(4\) in \(A\), write \(4 ∈ A\).

If a number is not contained in a set, for example the \(5\) in \(A\), write \(5 ∉ A\).

A set is defined by the different numbers contained in the set.
does not depend on the order and repetitions of the elements.

The two sets \(\{0,1,4,9\}\) and \(\{0,0,4,0,4,4,9,0,1,4,0,9\}\) are equal.

Two sets are equal if they contain the same elements.

A set can be empty. Remember the fruit basket mentioned above.
An empty set is represented by the symbol \(Ø\).

A set that is contained in another set is called a subset.

\(\{1, 4\}\) is a subset of \(\{0, 1, 4, 9\}\)

Average and union of sets

If you compare two subsets of a larger set, you can receive both a set that consists
of elements of both sets and, on the other hand,
a set that consists of elements that are contained in both sets.

A set consisting of elements of two subsets is called a union of two sets.

A set consisting only of elements contained in both sets is called the average of two sets.

Example of an union

Union of the sets \(A = \{0, 1, 4, 9\}\) und \(B = \{2, 5, 9\}\)

Write \(A ∪ B = \{1, 2, 4, 5, 9\}\)

In RedCrab Calculator use the function \(Union (A, B) = 1, 2, 4, 5, 9\)

Example of an average

Average of the sets \(A = \{0, 1, 4, 9\}\) und \(B = \{2, 5, 9\}\)

Write \(A ∩ B = \{9\}\)

In RedCrab Calculator use the function \(Intersect (A, B) = 9\)

Set difference is a subset obtained by comparing and subtracting two subsets.
The result consists only of elements of the first sets that are not contained in the second set

Difference of the sets \(A = \{0, 1, 4, 9\}\) and \(B = \{2, 5, 9\}\)

Write \(A\) \ \( B = \{0, 1, 4\}\)

In RedCrab Calculator use the function SetDiff \( (A, B) = 0, 1, 4\).

Complement

A complement is a subset obtained by comparing a subset \(A\) with the total set \(M\).
The result consists of only elements of M that are not contained in the subset \(A\).

Complement of the total set \(M = \{0,1,2,3,4,5,6,7,8,9,10\}\) and the subset \(A = \{0, 1, 4, 9\}\)

Write \(M / A = \{2, 3 ,5 ,6 ,7 , 8, 10\}\)

In RedCrab Calculator use the function \(SetDiff (M, A) = 2, 3 ,5 ,6 ,7 , 8, 10\)