Let a relation $ R $ in the set $N $ of natural numbers be defined by $(x, y)\Leftrightarrow x^2 - 4xy + 3y^2 = 0\,\forall\,x, y \in\,N.$The relation $ R$ is
- reflexive
- symmetric
- transitive
- an equivalence relation
The Correct Option is A
Solution and Explanation
Given that;
R is a relation on N defined by;
\(xRy=x^2-4xy+3y^2=0\)
⇒\((x-y)(x-3y)=0-(i)\)
The given equation is reflexive if x=a and y=a
i.e., \((a,a)\in R\forall N\) and symmetric
\((1,3) \) satisfies the equation (i)
\((1,3)\in R\)
\((1-3)(1-3\times3)\)
\(=16\neq 0\)
\(\therefore (1,3)\not\in R\)
\(\therefore \) Not symmetric
Let us assume the function is transitive for (9,1)
Let's Substitute and test \(\therefore (9-1)(9-3\times1)\)
\(=48\neq0\)
Hence, not transitive \(\therefore (9,1)\not\in R\)
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Concepts Used:
Functions
A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. Let A & B be any two non-empty sets, mapping from A to B will be a function only when every element in set A has one end only one image in set B.
Kinds of Functions
The different types of functions are -
One to One Function: When elements of set A have a separate component of set B, we can determine that it is a one-to-one function. Besides, you can also call it injective.
Many to One Function: As the name suggests, here more than two elements in set A are mapped with one element in set B.
Moreover, if it happens that all the elements in set B have pre-images in set A, it is called an onto function or surjective function.
Also, if a function is both one-to-one and onto function, it is known as a bijective. This means, that all the elements of A are mapped with separate elements in B, and A holds a pre-image of elements of B.
Read More: Relations and Functions