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Set Theory
Sets Explained👉Watch here
This week, our teacher covered the basics of the number system. We were instructed to consider 0 as part of the natural numbers, as it will be treated as such in future subjects like Python. However, in exams, it will be explicitly stated whether 0 should be considered a natural number.
The key topics from this week include set theory and the relationship between two sets. In set theory, we focused on three Venn diagram problems. In the context of relations, we discussed the concepts of reflexive, symmetric, transitive, and equivalence relations.
Detailed Explanation
1.Union of Two Sets
The union of two sets A and B is the set of elements that are in either A, B, or both. It is denoted as A ∪ B.
2.Intersection of Two Sets
The intersection of two sets A and B is the set of elements that are in both A and B. It is denoted as A ∩ B.
3.Subtraction of Two Sets
The subtraction of set B from set A (also known as the difference of sets) is the set of elements that are in A but not in B. It is denoted as A - B.
4.Union of Three Sets
The union of three sets A, B, and C is the set of elements that are in either A, B, C, or any combination of these sets. It is denoted as A ∪ B ∪ C.
Three Venn Diagram Problem Explained👉Watch here
5.Formulas
- Union of Two Sets: A ∪ B = { x | x ∈ A or x ∈ B }
- Intersection of Two Sets: A ∩ B = { x | x ∈ A and x ∈ B }
- Subtraction of Two Sets: A - B = { x | x ∈ A and x ∉ B }
- Union of Three Sets: A ∪ B ∪ C = { x | x ∈ A or x ∈ B or x ∈ C }
Relations
Relations Explained👉Watch here
A relation between two sets is a collection of ordered pairs containing one object from each set. If the object a is from set A and object b is from set B, then the pair (a, b) is a relation.
1.Reflexive Relation
A relation R on a set A is reflexive if every element is related to itself. Formally, ∀a ∈ A, (a, a) ∈ R.
2.Symmetric Relation
A relation R on a set A is symmetric if for all a and b in A, if (a, b) ∈ R, then (b, a) ∈ R.
3.Transitive Relation
A relation R on a set A is transitive if for all a, b, and c in A, if (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R.
4.Equivalence Relation
A relation R on a set A is an equivalence relation if it is reflexive, symmetric, and transitive.
5.Identity Relation
An identity relation on a set A is a relation where every element is related only to itself. Formally, I = { (a, a) | a ∈ A }.
Note: Identity is a type of reflexive relation which is always an equivalence relation.
6.Difference Between Identity and Reflexive Relations
While both identity and reflexive relations ensure that every element is related to itself, an identity relation is stricter. In an identity relation, each element is related only to itself, whereas in a reflexive relation, elements can be related to other elements as well.
Proof: Identity is Always an Equivalence Relation
To prove that an identity relation is always an equivalence relation, we need to show that it is reflexive, symmetric, and transitive:
- Reflexive: By definition, every element is related to itself in an identity relation.
- Symmetric: If (a, a) is in the relation, then (a, a) is trivially symmetric.
- Transitive: If (a, a) and (a, a) are in the relation, then (a, a) is also in the relation, satisfying transitivity.
7.Antisymmetric Relation
A relation R on a set A is antisymmetric if for all a and b in A, if (a, b) ∈ R and (b, a) ∈ R, then a = b.
8.Antitransitive Relation
A relation R on a set A is antitransitive if for all a, b, and c in A, if (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∉ R.
Function Explained in detailed👉Watch here
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