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Solution

Given,

$R_1=\{(1,1),(2,2),(3,3)\}$

$R_2=\{(1,1),(1,2),(1,3),(1,4)\}$

Now,

$R_1\cup R_2$

means all elements of $R_1$ and $R_2$.

So,

$R_1\cup R_2=\{(1,1),(1,2),(1,3),(1,4),(2,2),(3,3)\}$

Therefore, (A) matches with I.

Now,

$R_1-R_2$

means elements that are in $R_1$ but not in $R_2$.

Common element is $(1,1)$.

So,

$R_1-R_2=\{(2,2),(3,3)\}$

Therefore, (B) matches with IV.

Now,

$R_1\cap R_2$

means common elements of $R_1$ and $R_2$.

So,

$R_1\cap R_2=\{(1,1)\}$

Therefore, (C) matches with II.

Now,

$R_2-R_1$

means elements that are in $R_2$ but not in $R_1$.

So,

$R_2-R_1=\{(1,2),(1,3),(1,4)\}$

Therefore, (D) matches with III.

Final matching is:

A-I, B-IV, C-II, D-III