Sets & Relations
Sets
1. Define Sets.
Unordered collection of distinct objects.
2. Define Objects.
Objects that make up a set are called elements or members.
3. Define Cardinality.
Cardinality is the numbers of elements in a set.
4. Explain the use-case of uppercase and lowercase variables for denotation.
Uppercase to denote sets. Lowercase to denote elements of a set
5. Explain set membership and provide mathematical expression to represent elements in and not in the set $S$.
- $a \in S.$ “Element $a$ is in set $S$
- $d \notin S.$ “Element $d$ is not in set $S$
6. Explain subset and strict subset
- Subset:
- Set A is subset of Set B when all elements in Set A is also in Set B.
- $A \subseteq B$ (possibility of A=B)
- Strict Subset:
- Set A is subset of Set B when all elements in Set A is also in Set B and Set B cardinality is more than Set A
- $A \subset B$
7. Explain disjoint, universal and complement of set A
- Disjoint: No common elements between the sets.
- Universal: All encompassing set consisting of all the sets
- Complement of set A: All elements in universal set excluding a particular set A.
8. Explain numerically equivalent of sets.
- Both sets have the same cardinality.
- If both Set $A$ and $B$ has 3 elements then, $|A| = |B| =3$
9. Explain pairwise disjoint set.
A collection of sets in which every pair of sets within the collection has no elements in common.
10. Explain cartesian product of two sets $A \times B$ . and the cardinality of the resulting set.
- The cartesian product is mathematical operation that produces a new set consisting of all possible ordered pairs where the first element of each pair comes from set A and the second element comes from set B.
- The cardinality of the resulting set is the length of $A \times B$
11. Provide the cartesian product of $A = {1,2}, B= {x,y}$
$A \times B = {(1,x), (1,y), (2,x), (2,y)}$
12. Explain countable set (Countability infinite) and provide an example.
- A countable set is a set that has the same cardinality (size) as the set of natural numbers (positive integers)
- Example**:** The Set of Natural Numbers (N):
13. Explain uncountable set and provide an example.
- An uncountable set is a set that is not countable. In other words, its elements cannot be put into a one-to-one correspondence with the natural numbers.
- Example: Set of Real Numbers (ℝ): The set of real numbers, which includes all rational and irrational numbers, is uncountable.
Relations
1. Explain relations intuitively.
Relations establishing a connection between two objects. A relation describes the relationship between two objects and they are usually represented as an ordered pairs from one or more sets.
2. Explain the domain and range in relations.
- Domain and Range: In a relation, the domain is the set of all first components of the ordered pairs, and the range is the set of all second components of the ordered pairs.
- The domain and range provide information about which elements are involved in the relation.
3. What is the use-case of relations in math.
Relations facilitate the mapping of elements from a set, known as the domain, to elements in another set, referred to as the range. This mapping results in ordered pairs in the form (input, output).
4. Find the inverse of $R = {(1,2), (3,10)}$
$R^{-1} = {(y,x):(x:y)\in R }$
$R^{-1} = {(2,1), (10,3) }$