A set is a collection of objects called elements. They can be both finite and infinite.
A finite set will look something like this,
A = {1, 2, 3, 4, 5, 6, 7, 8, 9}
An infinite set will be like,
Z = {1, 2, 3, 4, …}
Coming to subsets, suppose if A is a subset of B then every element of A must also be in B. It is usually a set of positive integers that start from 1 to infinity. The dots indicate an implied pattern that goes on forever.
These are the Fundamental Notions of Sets.
Now let us consider some additional points–
- The repeated elements are only listed once.
Example –
{a, b, c, a, b, c} = {a, b, c}
Here the amount of numbers is not that important as all of them can be drawn into a single circle and only thing matter is what elements are written inside the circle.
- There is also no order in a set.
Example –
{3, 2, 1} = {1, 2, 3} = {2, 1, 3}
Next important subtopic is that of Common Sets
- Natural numbers – there can be two ways to write a natural number common set,
N = {0, 1, 2, 3, …}  or, N = {1, 2, 3, 4, ….}
Mostly experts recommend using the set with natural number starting from 1 as these kinds of sets are more specific due to having Z+ positive integer.
- Integers –
These are a set of numbers which are either positive or negative and also whole numbers so they can go from negative infinity till positive infinity.
Example –
Z= {…, -2, -1, 0, 1, 2, …}
This is one fundamental notion that students needs to use a lot in discrete math.
- Rational numbers –
It is kind of hard to list and mainly contains fractional numbers in a set.
Example –
Q = {1/1, ½, 1/3, 2/3, …}
Hence, rational numbers are any numbers that can be written as fractions.
This knowledge helps to find solution of the problem how many solutions are there to the equation x1+x2+x3+x4+x5=21?
For which the answer should be 106.
Now that we know what sets are along with the additional points about it, coming to the next topic,
Elements and cardinality
It talks about the things present in the sets and how big those sets are.
Sets need to be all mathematics, it can be words too that have meaning in the real world.
Example –
- “Yellow is an element of C†– yellow is inside the set C.
This is usually represented as yellow ∈ C
- “Green is not in the set C†– green is not present inside the set C.
This is represented as green ∉ C.
- The size of C is 3 which mean the amount of things in C is 3. There are 3 different elements in C. Here the absolute value or the size of c is equal to 3.
This is symbolized as- lcl = 3
Which says that the cardinality of C is 3?
Knowing the answer of thee question how many solutions are there to the equation x1+x2+x3+x4+x5=21 cannot be impossible if one do not have the basic knowledge of discrete math.