Let R be the relation defined in 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 and all the elements of the subset {2,4,6} are related to each other, but no element of the subset {1,3,5,7} is related to any element of the subset {2,4,6}. Solution Given any element a in A, both a and a must be either odd or even, so that (a,a)∈R. Further, (a,b)∈R⇒ both a and b must be either odd or even ⇒(b,a)∈R. Similarly, (a,b)∈R and (b,c)∈R⇒ all elements a, b, c, must be either even or odd simultaneously ⇒(a,c)∈R. Hence, R is an equivalence relation. Further, all the elements of {1,3,5,7} are related to each other, as all the elements of this subset are odd. Similarly, all the elements of the subset {2,4,6} are related to each other, as all of them are even. Also, no element of the subset {1,3,5,7} can be related to any element of {2,4,6}, as elements of {1,3,5,7} are odd, while elements of {2,4,6} are even.
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Step-by-Step Solution
Step 1: Check for Reflexivity
A relation R is reflexive if every element is related to itself. For any element a in set A, both a and a are either odd or even (they are the same number). Therefore, (a, a) satisfies the condition for relation R, meaning R is reflexive.
Step 2: Check for Symmetry
A relation R is symmetric if whenever (a, b) is in R, then (b, a) is also in R. If (a,b)∈R, it means both a and b are either odd or even. This condition is symmetrical, meaning if a and b are both odd/even, then b and a are also both odd/even. Hence, (b,a)∈R, and R is symmetric.
Step 3: Check for Transitivity
A relation R is transitive if whenever (a, b) is in R and (b, c) is in R, then (a, c) is also in R. If (a,b)∈R, then a and b are both odd or both even. If (b,c)∈R, then b and c are both odd or both even. Combining these, it implies that a, b, and c must all be either odd or all even. Therefore, (a, c) must also satisfy the condition (both a and c are either odd or even), so (a,c)∈R. Thus, R is transitive.
Step 4: Conclusion for Equivalence Relation
Since the relation R satisfies all three properties: reflexivity, symmetry, and transitivity, it is an equivalence relation.
Step 5: Analyze Subsets
Consider the subset S1={1,3,5,7}. All elements in S1 are odd. According to the definition of R, any two elements from S1 will be related to each other because they are both odd. Similarly, for the subset S2={2,4,6}, all elements are even. Thus, any two elements from S2 will be related to each other because they are both even.
Step 6: Analyze Relation Between Subsets
If we take an element a from S1 (which is odd) and an element b from S2 (which is even), then a and b are not both odd or both even. One is odd and the other is even. Therefore, (a, b) does not satisfy the condition for relation R, meaning no element from S1 is related to any element from S2.