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Let $X=\{1,2,3\}$ and $Y=\{4,5\}$. Find whether the following subsets of $\mathrm{X} \times \mathrm{Y}$ are functions from $\mathrm{X}$ to $\mathrm{Y}$ or not.
(i) $\mathrm{f}=\{(1,4),(1,5),(2,4),(3,5)\}$
(ii) $g=\{(1,4),(2,4),(3,4)\}$
(iii) $\mathrm{h}=\{(1,4),(2,5),(3,5)\}$
(iv) $\mathrm{k}=\{(1,4),(2,5)\}$
(i) $\mathrm{f}=\{(1,4),(1,5),(2,4),(3,5)\}$
(ii) $g=\{(1,4),(2,4),(3,4)\}$
(iii) $\mathrm{h}=\{(1,4),(2,5),(3,5)\}$
(iv) $\mathrm{k}=\{(1,4),(2,5)\}$
Solution:
1686 Upvotes
Verified Answer
Given that, $X=\{1,2,3\}$ and $Y=\{4,5\}$
$$
X \times Y=\{(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)\}
$$
(i) $\mathrm{f}=\{(1,4),(1,5),(2,4),(3,5)\}$
$\mathrm{f}$ is not a function because $\mathrm{f}$ does not have a unique image.
(ii) Since $g=\{(1,4),(2,4),(3,4)\}$
$\mathrm{g}$ is a function as each element of the domain has unique image.
(iii) $\mathrm{h}=\{(1,4),(2,5),(3,5)\}$
It is clear that $h$ is a function.
(iv) $\mathrm{k}=\{(1,4),(2,5)\}$
$\mathrm{k}$ is not a function as 3 has not any image under the mapping.
$$
X \times Y=\{(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)\}
$$
(i) $\mathrm{f}=\{(1,4),(1,5),(2,4),(3,5)\}$
$\mathrm{f}$ is not a function because $\mathrm{f}$ does not have a unique image.
(ii) Since $g=\{(1,4),(2,4),(3,4)\}$
$\mathrm{g}$ is a function as each element of the domain has unique image.
(iii) $\mathrm{h}=\{(1,4),(2,5),(3,5)\}$
It is clear that $h$ is a function.
(iv) $\mathrm{k}=\{(1,4),(2,5)\}$
$\mathrm{k}$ is not a function as 3 has not any image under the mapping.
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