Prove that the relation R in the set A=1,2,3,4,5 given by R=(a,b):∣a−b∣iseven is an equivalence relation.
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Step-by-Step Solution
Step 1: Understand Equivalence Relation
An equivalence relation partitions a set into disjoint subsets where all elements within a subset are related to each other. To prove that a given relation is an equivalence relation, we must demonstrate that it satisfies three fundamental properties: reflexivity, symmetry, and transitivity.
Step 2: Check for Reflexivity
For a relation to be reflexive, every element must be related to itself. In this case, we need to check if |a - a| is even for any element a in the set A. The absolute difference between an element and itself is always 0.
Step 3: Evaluate Reflexivity Condition
Since 0 is an even number (as it can be written as 2×0), the condition |a - a| is even is satisfied for all a∈A. Therefore, the relation R is reflexive.
Step 4: Check for Symmetry
For symmetry, if an element a is related to an element b, then b must also be related to a. We assume that (a,b)∈R, which means that the absolute difference between a and b is an even number.
Step 5: Evaluate Symmetry Condition
The absolute value of a number is the same as the absolute value of its negative. Therefore, |a - b| is equal to |b - a|. Since we assumed |a - b| is even, it follows that |b - a| must also be even. This implies that (b,a)∈R, so the relation R is symmetric.
Step 6: Check for Transitivity
For transitivity, if a is related to b, and b is related to c, then a must be related to c. We assume that (a,b)∈R and (b,c)∈R. This means that a - b and b - c are both even integers (positive or negative, as their absolute values are even).
Step 7: Express differences as even integers
If the absolute value of a difference is even, then the difference itself must be an even integer (it could be positive or negative). So, a-b can be written as an even integer 2m and b-c can be written as an even integer 2n, where m and n are integers.
Step 8: Find the difference between a and c
To check if (a,c)∈R, we need to evaluate the difference a - c. We can express a - c as a sum of the differences a - b and b - c, which we already know are even integers.
Step 9: Substitute and sum the even integers
Substituting the representations for a - b and b - c as 2m and 2n respectively, we find that their sum 2m + 2n simplifies to 2(m + n). Since m and n are integers, their sum (m + n) is also an integer. This means that a - c is twice an integer.
Step 10: Evaluate Transitivity Condition
Since a - c can be expressed as 2×an integer, it is an even integer. Therefore, its absolute value |a - c| is also even. This proves that (a,c)∈R, and thus the relation R is transitive.
Step 11: Conclusion
Since the relation R satisfies all three properties required for an equivalence relation (reflexivity, symmetry, and transitivity), we can conclude that R is an equivalence relation on the set A.