Sets

A set is a collection of well-defined and distinct objects, known as elements.
The symbol to denote a set is a pair of curly braces {} – for instance, {1, 2, 3} is the set of integers 1, 2, and 3.
Elements within a single set are unique - it means if a set is defined as {1, 2, 3, 2}, it is effectively the same as the set {1, 2, 3}.
Elements in a set do not have a specific order. Hence, {1, 2, 3} is identical to the set {3, 2, 1}.
Sets can also have unlimited numbers, such as the set of all positive integers.
If a set has no element it is called an empty set or null set and denoted as {} or Ø.

Set notation is a language used to precisely describe the contents of a set. A set can be described in roster form (by listing all elements, e.g., {1,2,3}), or in set-builder form (a rule that states the property or properties that elements must have, e.g., {x

x > 0}), or by context (e.g., the set of vowels in the English alphabet).

The universe of discourse or universal set is the set containing all objects under consideration for a given discussion or problem.
A subset is a set, every member of which is a member of another set. We use the ‘⊆’ symbol to denote a subset. For example, if A = {1, 2, 3} and B = {1, 2, 3, 4, 5}, then A is a subset of B.
An intersection of two or more sets is a new set containing only the elements that are common to all the sets. The symbol for intersection is ‘∩’. For instance, if A = {1, 2, 3} and B = {2, 3, 4}, then A ∩ B = {2, 3}.
A union of sets is a set containing all elements that are in any of the sets. The symbol for union is ‘∪’. For instance, if A = {1, 2, 3} and B = {2, 3, 4}, A ∪ B = {1, 2, 3, 4}.
Two sets are said to be disjoint if they have no elements in common. Therefore, the intersection of disjoint sets is an empty set.
The complement of a set A, denoted by A’, consists of all elements of the Universe set U that are not in A.
With the help of Venn diagrams, you can graphically display how much different groups have in common. The universal set is usually displayed as rectangle, subsets as circles within the rectangle, and the common elements of the sets are represented by the areas of overlap among the circles.
You can find the number of elements in the union of two sets using this formula: n(A ∪ B) = n(A) + n(B) – n(A ∩ B).
This subject often overlaps with probability theory — for example, you might need to find the probability that an event drawn from a universal set belongs to a certain subset.
Problems can be solved using a combination of set theory principles. Practice is key for mastery, so keep working on problems until you feel comfortable with all concepts.

Remember, understanding the principles of sets and being able to use them effectively is an important part of getting a full grasp of number systems.