Sets and Subsets

Sets and Subsets are two important concepts in mathematics that are taught in middle school and are also used in practical mathematics. Sets are collection of elements such as numbers, letters, things, etc. For example: { kids wooden puzzles , kids’ building block, kids’ construction toys}. Here is a set of educational toys such as kids’ wooden puzzles and so on. Sets can be finite, infinite, singleton and empty sets.

Subsets:
A set is defined by its elements. If two sets have same elements, they are referred as equal sets. On the other hand if a set is empty, it is referred as empty set. Subsets too are defined by the elements of sets. If A and B are two sets, A can be defined as a subset of B only when every element of A is also an element of B.

For example:
A = {remote helicopter toy, Barbie doll, Video game , electronic toy}
B = {Construction toy, Puzzles, Dart board, remote helicopter toy , Barbie doll, Video game, electronic toy}
Here, A is a subset of B.
Power Set:
A power set is a set of all the subsets of a set.

For example:
A = {Apple, Mango, Banana, Orange, Grapes, Papaya, Jack fruit, Blue berry}
Now, {Apple}, {Mango, Banana}, {Orange}, {Grapes}, {Papaya, Jack fruit, Blue berry} are subsets of A
{Apple, Mango, Banana, Orange, Grapes, Papaya, Jack fruit, Blue berry} is subset of A
{} is an empty set and it is also a subset of A
Now, the list of all the subsets is called the power set. Power set is represented by P(S).
Therefore, P(S) of A = {{Apple}, {Mango, Banana}, {Orange}, {Grapes}, {Papaya, Jack fruit, Blue berry}, {Apple, Mango, Banana, Orange, Grapes, Papaya, Jack fruit, Blue berry} {}}
If a set has n number of elements, the power set will have 2n number of elements. For example: A = (a,b,c). The number of elements in the power set will be 23 = 8 elements.
These are the basics of sets, subsets and power set.