Page 13 - Mathematics Class - XII

DEMONSTRATION

1. Let the threads or yarns represent the lines l , l , ... l .
1
2
8
2. l and l are perpendicular to each of the lines l , l , l and l . [See Fig. (c)]
1 2 3 4 5 6
3. l is parallel to l .
7 8
4. l is parallel to l , l is parallel to l and l is parallel to l .
6 3 3 4 5 6
5. (l , l ), (l , l ), (l , l ), (l , l ), (l , l ),...∈ R
1 2 3 4 3 5 5 6 7 8

OBSERVATION
1. We see that each line is parallel to itself,
So, the relation

R = {(l, m): l || m} is reflexive.

2. Observe lines l and l , we see that l || l .
1
2
2
1
Similarly, l || l
2 1
∴ (l , l ) ∈ R ⇒ (l , l ) ∈ R
1 2 2 1
Similarly, we see that l || l ⇒ l || l
3 4 4 3
∴ (l , l ) ∈ R ⇒ (l , l ) ∈ R
3 4 4 3
Also, lines l || l ⇒ l || l 7
8
7
8
∴ (l , l ) ∈ R ⇒ (l , l ) ∈ R
7 8 8 7
∴ The relation R is symmetric.
3. We see lines (l and l ) and (l and l ), we see that l || l and l || l ⇒ l || l .
3 4 4 5 3 4 4 5 3 5
∴ (l , l ) ∈ R and (l , l ) ∈ R ⇒ (l , l ) ∈ R.
3 4 4 5 3 5
Thus, the relation R is transitive.

Hence, the relation R = {(l, m): l || m} is reflexive, symmetric and transitive. So, the relation R in this case is an
equivalence relation.

CONCLUSION
This activity is useful in understanding the concept of an equivalence relation.

APPLICATION
This activity can be repeated by taking some more threads or yarns in different positions.

Knowledge Booster
Equality (=) is an equivalence relation. It is an important relation, but is not a
very interesting example, since no two distinct objects are related by equality.

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