CameraIcon
CameraIcon
SearchIcon
MyQuestionIcon


Question

Let A={x Z:0x12}, R={(a,b):a,bA,|ab| is divisible by 4}, then which among the following options are correct


A
equivalence class of 1 is {1,5,9}
loader
B
R is reflexive and symmetric but not transitive relation.
loader
C
R is a Transitive relation
loader
D
equivalence class of 1 is A
loader

Solution

The correct options are
A equivalence class of 1 is {1,5,9}
C R is a Transitive relation
We have R={(a,b):a,bA,|ab| is divisible by 4}, where a,b{0,1,2,3,....,12}.

For any a A, we have
|aa|=0, which is divisible by 4
(a,a)R.
So, R is a reflexive relation.

Let (a,b)R
|ab|=4λ for some λZ
|ba|=4λ1 for some λ1Z  [|ab|=|ba|]
(b,a)R
So, R is a symmetric relation.

Let (a,b) and (b,c)R
ab=4α,bc=4β,   α,βZ
Now ac=4(α+β)
|ac| is divisible by 4
(a,c)R 
So, R is a transitive relation.
Hence, R is an equivalence relation.

Equivalance class of 1 is all possible values  of x such that (x,1)R,xA  |x1| is divisible by 4
|x1|=0,4,8
x1=0,4,8
x=1,5,9
Thus, elements related to 1  are {1,5,9}.

 

Mathematics

Suggest Corrections
thumbs-up
 
0


similar_icon
Similar questions
View More


similar_icon
People also searched for
View More



footer-image