Let $A = \{1, 2, 3, 4, 5\} \times \{1, 2, 3, 4, 5\}$ and define $R$ on $A$ by $(X_1,Y_1)R(X_2,Y_2)$ if $X_1+Y_1 = X_2+Y_2$.
(a) Verify that $R$ is an equivalence relation on $A$
(b) Determine the equivalence classes $[(1,3)], [(2,4)],$ and $ [(1,1)]$
(c) Determine the partition induced by $R$
I don't understand how can I prove the equivalence relation on (a) Please help
Well, first try to figure out what the relationship means.
What does $(a,b)$ being related to $(c,d)$ means. Is $(2,4)$ related to $(1,2)$? If not what is related to $(2,4)$
So $(a,b)$ being related to $(c,d)$ means $a+b = c+d$. And $(2,4)$ being related to $(1,2)$ would mean that $2+4 = 1+2$ which isn't true so $(2,4)$ is not related to $(1,2)$. So what is related to $(2,4)$. Well, $(2,4)$ is related to $(c,d)$ if $2+4=6= c+d$ so $(2,4)$ is related to $(c,d)$ if $c+d = 6$ so $(2,4)$ is related to $(1,5)$ and $(2,4)$ and $(3,3)$ and $(4,2)$ and $(5,1)$.
You should do all that before you start trying to do the question.
Then....
Go through the three conditions for equivalency.
1) is $R$ reflexive? Is it true that for any $(a,b)$ that $(a,b)$ $R$ $(a,b)$. Well, that's true if $a+b$ always equals $a+b$ and false if it doesn't always equal. As $a+b = a+b$ always, $R$ is relflexive.
2) is $R$ symmetric? Is it true that if $(a,b)\ R\ (c,d)$ then it must be true that $(c,d)\ R\ (a,b)$? Well, that's true if whenever $a+b = c+d$ that it must be true that $c+d =a+b$. Equality is equality and you can switch the order.
3) is $R$ transitive? Is it true that $(a,b)R(c,d)$ and $(c,d)R(e,f)$ that it must be true that $(a,b)R(c,d)$? Well that's true if whenever you have $a+b = c+d$ and $c+d=e+f$ then you must have $a+b = e+f$. Well, equality is equality and equality is transitive.
In general this is a relation based on view if sums are equal and equality is the most basic of equavalence relations so...
As a bonus to get you familiar with the idea of equivalence relations, hear are all $25$ elements of $A\times A$ listed into equivalence classes. Two pairs are equivalent if and only if the fall in the same class.
Class 1: Those that add to $2$: $\{(1,1)\}$. $(1,1)$ is only related to itself.
Class 2: Those that add to $3$: $\{(1,2), (2,1)\}$. Those in this class are all related to each other.
Class 3: Those that add to $4$: $\{(1,3), (2,2),(3,1)\}$. Those in this class are all related to each other.
Class 4: Those that add to $5$: $\{(1,4), (2,3),(3,2),(4,1)\}$. Those in this class are all related to each other.
Class 5: Those that add to $6$: $\{(1,5), (2,4),(3,3),(4,2), (5,1)\}$. Those in this class are all related to each other.
Class 6: Those that add to $7$: $\{(2,5), (3,4),(4,3),(5,2)\}$. Those in this class are all related to each other.
Class 7: Those that add to $8$: $\{(3,5), (4,4),(5,3)\}$. Those in this class are all related to each other.
Class 8: Those that add to $9$: $\{(4,5), (5,4)\}$. Those in this class are all related to each other.
Class 9: Those that add to $10$: $\{(5,5)\}$. $(5,5)$ is only related to itself.
So question b) is $[(1,3)]$ means the class of all the pairs that are reelated to $(1,3)$. Which of the above classes is that.
And question c) is... well, reread your text to see what that means and see if it makes sense now.