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Sets and Relations question

2023 · 24 Jan · Shift 2 · Q37
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  5. /2023 · 24 Jan · Shift 2 · Q37

Sets and Relations question

2023 · 24 Jan · Shift 2 · Q37

JEE MainMathematicsSets and RelationsNumerical+4 / −1
The minimum number of elements that must be added to the relation R = {(a, b), (b, c), (b, d)} on the set {a, b, c, d} so that it is an equivalence relation, is ‾\underline{\hspace{2cm}}​.
Numerical answer
View written solutionFree

Correct answer: 13

  1. Given relation

We have a set A={a,b,c,d}A=\{a,b,c,d\}A={a,b,c,d} and a relation R={(a,b),(b,c),(b,d)}.R=\{(a,b),(b,c),(b,d)\}.R={(a,b),(b,c),(b,d)}.

We must add the minimum number of ordered pairs so that the resulting relation becomes an equivalence relation.

An equivalence relation must be:

  1. Reflexive
  2. Symmetric
  3. Transitive
  1. Think in terms of equivalence classes

Since (a,b)∈R(a,b)\in R(a,b)∈R, in any equivalence relation we must have a∼ba\sim ba∼b.

Also, (b,c)∈R(b,c)\in R(b,c)∈R implies b∼cb\sim cb∼c.

Also, (b,d)∈R(b,d)\in R(b,d)∈R implies b∼db\sim db∼d.

Now by transitivity of equivalence:

  • from a∼ba\sim ba∼b and b∼cb\sim cb∼c, we get a∼ca\sim ca∼c
  • from a∼ba\sim ba∼b and b∼db\sim db∼d, we get a∼da\sim da∼d
  • from c∼bc\sim bc∼b and b∼db\sim db∼d, we get c∼dc\sim dc∼d

Thus all four elements must belong to the same equivalence class.

So the smallest equivalence relation containing the given pairs is the universal equivalence relation on AAA, namely A×A.A\times A.A×A.

Since ∣A∣=4|A|=4∣A∣=4, we have ∣A×A∣=42=16.|A\times A|=4^2=16.∣A×A∣=42=16.

  1. Count how many pairs are already present

Initially, R={(a,b),(b,c),(b,d)}R=\{(a,b),(b,c),(b,d)\}R={(a,b),(b,c),(b,d)} contains exactly 333 ordered pairs.

Therefore, the minimum number of pairs to be added is 16−3=13.16-3=13.16−3=13.

  1. Verification via properties

If all 161616 pairs are present, then certainly:

  • Reflexive: (a,a),(b,b),(c,c),(d,d)(a,a),(b,b),(c,c),(d,d)(a,a),(b,b),(c,c),(d,d) are present
  • Symmetric: whenever (x,y)(x,y)(x,y) is present, so is (y,x)(y,x)(y,x)
  • Transitive: since every pair is present, transitivity holds automatically

Hence this is indeed an equivalence relation.

Also, no smaller equivalence relation is possible, because the given relation forces all four elements into one equivalence class.

  1. Final answer

The minimum number of elements to be added is 13.\boxed{13}.13​.

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