Mathematics Lab Activity-02 Class XII

Mathematics Laboratory Activities on Relations & Functions for class XI Non-Medical students with complete observation tables strictly according to the CBSE syllabus.

Chapter - 01 | relations & functions

Activity - 02

Objective

To verify that the relation R in the set L of all lines in a plane, defined by R = {(l, m) : l || m} is an equivalence relation.

Material Required

A piece of plywood, 8 pieces of wires, nails, white paper, glue etc 

Procedure

1) Take a piece of plywood and paste a white paper on it. Fix the wires randomly on the plywood with the help of nails such that some of them are parallel, some are perpendicular to each other and some are inclined as shown in the figure 2.1

Figure 2.1

2.) Let wires represents the lines l₁ , l₂ , l₃ , ………. l₈.

3.) Line l₁ , is perpendicular to each of the lines l₂ , l₃ , l₄.

4.) Line l₆ is perpendicular to l₇

5.) Line  l₂ is parallel to  l₃, l₃ is parallel to , l₄ .

6.) Pair of lines (l₂ , l₃), (l₃ , l₄), (l₅ , l₈)  ∈ R

Observations

1) Reflexivity:

 It is clear from the figure 2 that every line is parallel to itself.

⇒ (l₁|| l₁) So (l₂|| l₂), (l₃|| l₃) ………∈ R  

⇒ (a, a) ∉ R  ∀ a ∈ Real numbers

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

2) Symmetry: 

In fig. 2. we see that L₂ || l₃  and l₃ || l₂   ⇒ (l₂, l₃) ∈ R and (l₃, l₂) ∈ R

Similarly : l₃ || l₄  and l₄ || l₃   ⇒ (l₃, l₄) ∈ R and (l₄, l₃) ∈ R

  l₅ || l₈  and l₈ || l₅   ⇒ (l₅, l₈) ∈ R and (l₈, l₅) ∈ R

In all the cases we see that for all a, b ∈ Real number we have  

(a, b) ∈R   ⇒  (b, a) ∈ R   

⇒ The relation R = {(l, m) : l || m } is symmetric.

3) Transitivity:  

(l₂ , l₃) ∈ R ⇒ l₂ || l₃

(l₃ , l₄) ∈ R ⇒ l₃ || l₄

From the figure it is clear that l₂ || l₄ ⇒ (l₂ , l₄) ∈ R

Here we conclude that for all a, b, c ∈ R we have 

(a, b) ∈ R, (b, c) ∈ R but (a, c) ∈ R

The relation R = {(l, m) : l || m } is transitive.

Result

Given relation R is reflexive, symmetric and transitive hence R is an equivalence relation.

Note : If (a, b) ∈ R but if there no ordered pair whose first element is b even then the function is called transitive.

Applications

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

VIVA – VOICE

Q. 1 What is reflexive relation ?
Ans. If all elements of set A are related to itself, then the relation is said to be a reflexive relation.

Q. 2 What is a symmetric relation ?

Ans. A relation said to be symmetric relation if aRb = bRa ∀ (a, b) ∈ R

Q.3 What is a transitive relation ?

Ans. A relation is said to be transitive relation if aRb ∈ R and bRc ∈ R ⇒ aRc ∈ R, ∀ a, b ∈ R.

Q4. What is antisymmetric relation ?

A relation R on set A is said to be anti-symmetric relation if (a, b) ∈ R and (b, a)∈ R then a = b.

Q5. Define equivalence relation ?

Ans. A relation which is reflexive, symmetric and transitive is called an equivalence relation.

Q6. Is intersection of two equivalence relations is a reflexive relation ?

Ans. Yes

Q7. Is intersection of two equivalence relations is a symmetric relation ?

Ans. Yes

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