Give more examples for collection of objects
Answers
A set is a collection of objects. The objects are called the elements of the set. If a set has finitely many elements, it is a finite set, otherwise it is an infinite set.
If the number of elements in a set is not too many, we can just list them
out. We usually use capital letters to name sets and use braces {} to de-limit a
set:
Example: S = {1, 4, 10} is the set that contains three elements, namely the
numbers 1, 4, and 10.
When the size of the set is too large or infinite, we just give a description of
them without listing them. For example, the set of real numbers, the set of even
integers, the set of all books written before the year 2000.
If two sets A and B have the same elements, we say that they are equal, and
write A = B.
A subset of a set is a sub-collection of the set. For example, if S = {1, 4, 10},
then A = {1, 4} is a subset of S since A is a sub-collection of S. (We say that
every element of A is an element of S.) If A is a subset of S, we write A ⊆ S.
By definition, every set is a subset of itself. If A is a subset of S but A 6= S,
we say that A is a proper subset of S. If A is a proper subset of S, we write
A ⊂ S.
The set that contains no object is the empty set, and is denoted by the mathematical symbol ∅. The empty set is a subset of any set.
If A, B are sets, the union of the two sets, denoted by A ∪ B, is the (most likely
bigger) set whose elements are elements that are either in A or in B (or both).
If A, B are sets, the intersection of the two sets, denoted by A∩B, is the (most
likely smaller) set whose elements are elements that are in both sets.
Example: If A = {3, 6, 9, 11, 15, 17}, and B = {0, 3, 9, 10, 15}, then
A ∪ B = {0,3, 6, 9, 10, 11, 15, 17}
A ∩ B = {3, 9, 15}
If a set A is finite, we define the order of the set, denoted by |A|, to be the
number of elements in a set.
For example, for the above sets A, and B, since set A has 6 elements, so |A| = 6.
Also, |B| = 5. A ∪ B has 8 elements, therefore |A ∪ B| = 8, and |A ∩ B| = 3.
You may have noticed that, for the above example, the number of elements in
the union of A and B is equal to the number of elements in A plus the number
of elements in B, then subtract the number of elements in the intersection of A
and B.