A relationship between two elements of a set is called a binary relationship. A binary relationship is said to be in equivalence when it is reflexive, symmetric, and transitive. A binary relationship is a reflexive relationship if every element in a set S is linked to itself. In mathematical terms, it can be represented as (a, a) ∈ R ∀ a ∈ S (or) I ⊆ R. Here, a is an element, S is the set and R is the relation. I is the identity relation on A. For example, let us consider a set C = {7,9}. Here the reflexive relation will be R = {(7,7), (9,9), (7,9), (9,7)}. A set of real numbers is also a reflexive set, because each element i.e. each real number “is equal to” itself.
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According to the reflexive property, (a, a) ∈ R, for every a ∈ S, where a is an element, R is a relation and S is a set. For instance, let us assume that all positive integers are included in the set X. Now, for all pairs of positive integers in set X, ((p,q),(p,q))∈ R. Then, we can say that (p,q) = (p,q) for all positive integers.
This proves the reflexive property of equivalence.
Statement | Symbol |
“is equal to” (equality) | = |
“is a subset of” (set membership) | ⊆ |
“divides” (divisibility) | ÷ or / |
“is greater than or equal to” | ≥ |
“is less than or equal to” | ≤ |
In a given set there are a number of reflexive relations that are possible. Let us consider a set S. This set has an ordered pair (p, q). Now, p can be chosen in n number of ways and so can q. Therefore, this set of ordered pairs comprises of n2 pairs. As per the concept of a reflexive relationship, (p, p) must be included in such ordered pairs. Also, there will be a total of n pairs of such (p, p) pairs. As a result, the number of ordered pairs will be n2-n pairs. Hence, the total number of reflexive relationships in set S is \[2^{n(n-1)}\].
The formula for the number of reflexive relations in a given set is written as N = \[2^{n(n-1)}\].
Here, N is the total number of reflexive relations, and n is the number of elements.
Some of the characteristics of a reflexive relation are listed below: -
Anti - Reflexive: If the elements of the set do not relate to themselves, they are said to be irreflexive or anti-reflexive.
Quasi - Reflexive: If each element that is related to a specific component, which is also related to itself, then that relationship is called quasi-reflexive. If a set A is quasi-reflexive, this can be mathematically represented as: ∀ a, b ∈ A: a ~ b ⇒ (a ~ a ∧ b ~ b).
Co - Reflexive: The relationship ~ (similar to) is co-reflexive for all elements a and b in set A if a ~ b also implies that a = b.
It is impossible for a reflexive relationship on a non-empty set A to be anti-reflective, asymmetric, or anti-transitive.
Example 1: A relation R on set A (set of integers) is defined by “x R y if 5x + 9x is divisible by 7x” for all x, y ∈ A. Check if R is a reflexive relation on A.
Solution:
Consider x ∈ A.
Now, 5x + 9x = 14x, which is divisible by 7x.
Therefore, x R y holds for all the elements in set A.
Hence, R is a reflexive relationship.
Example 2: A relation R is defined on the set of all real numbers N by ‘a R b’ if |a-a| ≤ b, for a, b ∈ N. Show that the R is not a reflexive relation.
Solution:
N is a set of all real numbers. So, b =-2 ∈ N is possible.
Now |a – a| = 0. Zero is not equal to nor is it less than -2 (=b).
So, |a-a| ≤ b is false.
Therefore, the relation R is not reflexive.
Example 3: A relation R on the set S by “x R y if x – y is divisible by 5” for x, y ∈ A. Confirm that R is a reflexive relation on set A.
Solution:
Consider, x ∈ S.
Then x – x= 0. Zero is divisible by 5.
Since x R x holds for all the elements in set S, R is a reflexive relation.
Example 4: Consider the set A in which a relation R is defined by ‘m R n if and only if m + 3n is divisible by 4, for x, y ∈ A. Show that R is a reflexive relation on set W.
Solution:
Consider m ∈ W.
Then, m+3m=4m. 4m is divisible by 4.
Since x R x holds for all the elements in set W, R is a reflexive relation.
1. What is the Set Theory in Mathematics?
Mathematical set theory was invented for the first time by Georg Cantor in 1874. The definition of sets in mathematics deals with the properties and operations of arrays of objects. This is very important for classification, organisation and is the basis for many forms of data analysis. A special relationship that may or may not exist between an object and a set is called a membership relationship. An object is or is not a member of a set; there is no in-between object. Set theory is seen as an intellectual foundation on which almost all abstract mathematical theories can be derived.
2. Who Invented Sets?
Mathematical set theory was invented for the first time by Georg Cantor in 1874. He was a German mathematician. He first presented his theories on sets in a paper called "On the Characteristic Property of All Real Algebraic Numbers." Although there were a lot of abstract concepts in math, like infinity. Cantor has developed a more fundamental and rigid framework for these concepts.
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