Page 10 - Mathematics Class - XII

DEMONSTRATION

1. Let the threads/yarns represent the lines l , l ... l .
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2
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2. l is perpendicular to each of the lines l , l , l . [See Fig. (c)]
1 2 3 4
3. l is perpendicular to l .
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4. l is parallel to l , l is parallel to l and l is parallel to l .
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5. l ⊥ l , l ⊥ l , l ⊥ l , l ⊥ l , l ⊥ l and l ⊥ l .
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6. (l , l ), (l , l ), (l , l ), (l , l ), (l , l ), (l , l ),...∈ R.
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OBSERVATION
1. We see that no line is perpendicular to itself.
So, the relation, R = {(l, m): l ⊥ m} is not reflexive.
2. Observe lines l and l , we see that l ⊥ l .
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2
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Similarly, l ⊥ l
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∴ (l , l ) ∈ R ⇒ (l , l ) ∈ R
1 2 2 1
Similarly, we see that l ⊥ l ⇒ l ⊥ l
1 3 3 1
∴ (l , l ) ∈ R ⇒ (l , l ) ∈ R
1 3 3 1
Also, lines l ⊥ l ⇒ l ⊥ l 6
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∴ (l , l ) ∈ R ⇒ (l , l ) ∈ R
6 7 7 6
∴ The relation R is symmetric.
3. We see lines (l and l ) and (l and l ) as l ⊥ l and l ⊥ l but l ⊥ l .
1 2 2 5 1 2 2 5 1 5
∴ (l , l ) ∈ R and (l , l ) ∈ R but (l , l ) ∉R
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Hence, the relation R is not transitive.
CONCLUSION
It is verified that relation ‘is perpendicular to’ on the set R of all straight lines in a plane is symmetric but neither
reflexive nor transitive.
APPLICATION
This activity can be used to check whether a given relation is an equivalence relation or not.

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

Knowledge Booster
If there is only one output for every input, you have a function. If not, you have a
relation. Relations have more than one output for at least one input. Every function is
a relation. However, not every relation is a function.

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