Let N denote the set of all natural numbers. Define two binary relations on N as $R_1 = \{(x, y) \epsilon N \times N : 2x + y = 10 \}$ and $R_2 = \{(x, y) \epsilon N \times N : x + 2y = 10 \}$. Then :
- Range of $R_1$ is $\{2, 4, 8\}$
- Range of $R_2$ is $\{1, 2, 3, 4\}$
- Both $R_1$ and $R_2$ are symmetric relations
- Both $R_1$ and $R_2$ are transitive relations
The Correct Option is B
Solution and Explanation
$R_{1}=\{(x, y) \in N \times N: 2 x+y=10\}$ and $ R_{2}=\{(x, y) \in N \times N: x+2 y=10\} $
$R_{1}=\{(1,8),(2,6),(3,4),(4,2)\}$ and $ R_{2}=\{(8,1),(6,2),(4,3),(2,4)\}$
Therefore, range of $R_{2}=\{1,2,3,4\}$
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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