Let R be the relation defined on the set A={1,2,3,4,5,6,7} by R={(a,b): both a and b are either odd or even}. Show that R is an equivalence relation. Further, show that all the elements of the subset {1,3,5,7} are related to each other similarly all the elements of the subset {2,4,6} too.

Question

Let R be the relation defined on the set A={1,2,3,4,5,6,7} by R={(a,b): both a and b are either odd or even}. Show that R is an equivalence relation. Further, show that all the elements of the subset {1,3,5,7} are related to each other similarly all the elements of the subset {2,4,6} too.

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The relation R on set A={1,2,3,4,5,6,7} is defined by R={(a,b): both a and b are either odd or even}We observe the following properties of R on AReflexivity: Clearly, (1,1),(2,2),(3,3),(4,4),(5,5),(6,6),(7,7)∈R. So, R is a reflexive relation in ASymmetric: Let a,b∈A be such that (a,b)∈R. Then, (a,b)∈RBoth a and b are either odd or evenBoth b and a are either odd or even⇒ (b,a)∈RThus, (a,b)∈R⇒(b,a)∈R for all a,b∈ASo, R is a symmetric relation on ATransitivity: Let a,b,c∈Z be such that (a,b)∈R,(b,c)∈R. Then, (a,b)∈R⇒ Both a and b are either odd or even(b,c)∈R⇒ Both b and c are either odd or evenIf both a and b are even, then(b,c)∈R⇒ Both b and c are evenIf both a and b are odd, then(b,c)∈R⇒ Both b and c are odd∴ Both a and c are even or odd. Therefore (a,c)∈RSo, (a,b)∈R and (b,c)∈R⇒(a,c)∈RConsequently, R is a transitive relation on AHence, R is an equivalence relation on AWe observe that two numbers in A are related if both are odd or both are even. Since {1,3,5,7} has all odd numbers of A. So, all the numbers of {1,3,5,7} are related to each other. Similarly, all the numbers of {2,4,6} are related to each other as it contains all even numbers of set A. An even ,odd number in A is related to an even ,odd number in A respectively. So, no number of the subset {1,3,5,7} is related to any number of the subset {2,4,6}

Question Text	Let $R$ be the relation defined on the set $A = {1, 2, 3, 4, 5, 6, 7}$ by $R = {(a, b) : both a and b are either odd or even}$ . Show that $R$ is an equivalence relation. Further, show that all the elements of the subset ${1, 3, 5, 7}$ are related to each other similarly all the elements of the subset ${2, 4, 6}$ too.
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