Mathematics Lab Activity-1 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 - 1

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 symmetric but neither reflexive nor transitive.

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 1.1

Figure 1.1

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.) (l₁, l₂), (l₁ , l₃), (l₁ , l₄), (l₆ , l₇) ∈ R

Observations

1) Reflexivity:

Here we see that (l₁ , l₁) ∉ R

It is clear from the figure 1.1 that no line is perpendicular to itself.

⇒ (a, a) ∉ R ∀ a ∈ Real numbers

So the relation R = {(l, m) : l ⊥ m} is not reflexive.

2) Symmetry:  

It is clear from the figure  that l₁ ⊥ l₂  and l₂ ⊥ l₁  

So we have  (l₁ , l₂) ∈ R   ⇒   (l₂ , l₁) ∈ R

Similarly : l₁ ⊥ l₃  and l₃ ⊥ l₁   ⇒ (l₁ , l₃) ∈ R and (l₃ , l₁) ∈ R

 l₁ ⊥ l₄  and l₄ ⊥ l₁   ⇒ (l₁ , l₄) ∈ R and (l₄ , l₁) ∈ R

l₆ ⊥ l₇  and l₇ ⊥ l₆   ⇒ (l₆ , l₇) ∈ R and (l₇ ,l₆) ∈ R

In all the cases we see that for all a, b ∈ Real number we have
(a, b) ∈ R ⇒ (b, a) ∈ R
⇒ R = {(l, m) : l ⊥ m} is a symmetric relation.

3) Transitivity:

Here we see that l₂ ⊥ l₁ and l₁ ⊥ l₃ but l₂ is not ⊥ l₃

⇒ (l₂ , l₁) ∈ R and (l₁ , l₃) ∈ R but (l₂ , l₃) ∉ R

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

(a, b) ∈ R, (b, c) ∈ R but (a, c) ∉ R

⇒ R is not a Transitive relation.

Result

Given relation R is symmetric but neither reflexive nor transitive. Hence this relation is not 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 can be used to determine that whether a given relation is an equivalence relation or not.

VIVA – VOICE

Q. 1 What is identity relation ?
Ans. A relation I = {(a,a): a∈A} is said to be an identity relation.

Q. 2 What is empty relation ?
Ans. Let A and B are two non empty sets, then R is said to be an empty relation, if no element of A is related to any element of B.

Q. 3 What is universal relation ?
Ans. If all elements of set A are related to every element of set B, then it is said to be an universal relation.

Q. 4 Define a function ?
A function is a rule which associates for every x ∈ A if ∃ a unique y ∈ B such that f(x) = y.

Q. 5 Is every function a relation ?
Ans. Yes every function is a relation.

Q. 6 Is every relation a function ?
Ans. No, every function is not a relation.

Q. 7 How do we denote the cartesian product of two non empty sets ?
Ans. The cartesian product of two non empty sets A and B is denoted by A ✕ B.

Q. 8 Define domain of a relation ?

Set of all the first elements of all the ordered pair of a relation is called its domain.

Q. 9 Define range of a relation ?

Ans. Set of all the first elements of all the ordered pair of a relation is called its domain.

THANKS FOR YOUR VISIT

PLEASE COMMENT BELOW