138 views

Which of the relations below can also be characterized as a function defined on the set
I = { 1, 2, 3, 4, 5 }

1. { (x, y) | x, y ∈ I, x < y }
2. B{ (x, y) | x, y ∈ I, x = 1 }
3. C{ (x, y) | x, y ∈ I, x! = y }
4. None of these

A function from a set X to a set Y assigns each element of X to exactly one element of Y.

$I = {1,2,3,4,5}$

For $\{(x,y) | x,y \in I, x<y\}$, $(x,y)$ is in relation R if $x<y$

$\therefore \text{relation }R = \{(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)\}$, this relation is not a function because elements of domain are related to more than 1 element in co-domain.

For $\{(x,y) | x, y \in I, x=1\}$, $(x, y)$ is in relation $R$ if $x=1$

$\therefore \text{relation }R = \{(1,1),(1,2),(1,3),(1,4),(1,5)\}$, this relation is not a function because $1$ in domain is related to every element in co-domain and elements $2,3,4,5$ in domain are not related to anyone.

For $\{(x,y)|x,y \in I, x \neq y \}$, $(x,y)$ is in relation R if $x$ != $y$

$\therefore \text{realtion }R = \{(1,2),(1,3),(1,4),(1,5),(2,1),(2,3),(2,4),(2,5),(3,1),(3,2),(3,4),(3,5),(4,1),(4,2),(4,3),(4,5),(5,1),(5,2),(5,3),(5,4)$ clearly not a function because every element of domain is related to multiple elements in co-domain.

Hence, None of these is the answer.
by