Let X={1,2,3,4,5,6,7,8,9}. Let R1 be a relation in X given by R1={(x,y):x−y is divisible by 3 } and R2 be another relation on X given by R2={(x,y):{x,y}⊂{1,4,7}} or {x,y}⊂{2,5,8} or {x,y}⊂{3,6,9}}. Show that R1=R2.
Get the complete, step-by-step math solution for: "Let X=\{1{,}2{,}3{,}4{,}5{,}6{,}7{,}8{,}9\}. Let {R}_{1} be a relation in {X} given by {R}_{1}=\{(x, y): x-y is divisible by 3 \} and {R}_{2} be anoth...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Analyze Relation R1
Relation R1 states that x-y must be divisible by 3. This means that x and y must have the same remainder when divided by 3. We can categorize the elements of set X based on their remainders when divided by 3.
Step 2: Categorize Elements by Remainder (R1)
We divide the set X into three disjoint subsets based on their remainders when divided by 3. C0 contains numbers divisible by 3, C1 contains numbers with a remainder of 1, and C2 contains numbers with a remainder of 2.
Step 3: Express R1 using Categories
For x-y to be divisible by 3, both x and y must belong to the same category (C0, C1, or C2). This means that if (x,y)∈R1, then x and y must come from the same remainder class.
Step 4: Analyze Relation R2
Relation R2 is defined such that both x and y must belong to one of the specified subsets: {1,4,7}, {2,5,8}, or {3,6,9}.
Step 5: Compare R1 and R2
By comparing the definitions, we see that the subsets specified in R2 are exactly the remainder classes we identified for R1. Specifically, {1,4,7}=C1, {2,5,8}=C2, and {3,6,9}=C0.
Step 6: Conclusion
Since both relations define the same condition for (x,y) to be related (i.e., x and y must belong to the same remainder class modulo 3), we can conclude that R1=R2.