Question:
Define a relation $R$ on $A =\{1, 2, 3, 4\}$ as $_xR_y$ if $x$ divides $y$. $R$ is
Define a relation $R$ on $A =\{1, 2, 3, 4\}$ as $_xR_y$ if $x$ divides $y$. $R$ is
Updated On: Apr 18, 2024
- reflexive and transitive
- reflexive and symmetric
- symmetric and transitive
- equivalence
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The Correct Option is A
Solution and Explanation
Given set $A=\{1,2,3,4\}$ and relation, $x R y$ if $x$ divides $y$.
$\Rightarrow$ Relation
$=\{(1,1),(2,2),(3,3),(4,4),(1,2),(1,3)(1,4),(2,4)\}$
We have, $x R y \Leftrightarrow y / x$ for $x, y \in A$ For any $x \in A$, we have $x / x \Rightarrow x R x$
Thus, $x R x$ for all $x \in A .$ So, $R$ is reflexive on $A$.
$R$ ia not symmetry because, if $y / x$, then $x$ may not divide $y .$ For example $4 / 2$ but $2 / 4$
Let $x, y, z \in A$, such that $x R y$ and $y R z .$
Then, $x R y$ and $y R z \Rightarrow \frac{y}{x}$ and $\frac{z}{y} \Rightarrow \frac{z}{x} .$
So, $R$ is a transitive relation on $A$.
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Concepts Used:
Relations and functions
A relation R from a non-empty set B is a subset of the cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.
A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B. In other words, no two distinct elements of B have the same pre-image.
Representation of Relation and Function
Relations and functions can be represented in different forms such as arrow representation, algebraic form, set-builder form, graphically, roster form, and tabular form. Define a function f: A = {1, 2, 3} → B = {1, 4, 9} such that f(1) = 1, f(2) = 4, f(3) = 9. Now, represent this function in different forms.
- Set-builder form - {(x, y): f(x) = y2, x ∈ A, y ∈ B}
- Roster form - {(1, 1), (2, 4), (3, 9)}
- Arrow Representation
