wiz-icon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

Which of the following defined on Z is not an equivalence relation?


A

(x,y)Sxy

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
B

(x,y)Sx=y

No worries! We‘ve got your back. Try BYJU‘S free classes today!
C

(x,y)Sx-y is a multiple of 3

No worries! We‘ve got your back. Try BYJU‘S free classes today!
D

(x,y)S if x-y is even

No worries! We‘ve got your back. Try BYJU‘S free classes today!
Open in App
Solution

The correct option is A

(x,y)Sxy


Explanation for the correct answer

For option (a)

(x,y)Sxy

Reflexive: Let xS

xx is always hold then (x,x)S

Therefore, S is a reflexive relation.

Symmetric: Let x,yS

(x,y)Sxy it does not imply yx

Therefore, S is not a symmetric relation.

Hence, (x,y)Sxy is not an equivalence relation.

The explanation for incorrect options

For option (b)

(x,y)Sx=y

Reflexive: Let xS

x=x is always hold then (x,x)S

Therefore, S is a reflexive relation.

Symmetric: Let x,yS

x=yy=x(y,x)S

Therefore, S is a symmetric relation.

Transitive: Let x,y,zS

(x,y)Sx=y and (y,z)Sy=z

x=z(x,z)S

Therefore, S is a transitive relation.

Hence, (x,y)Sx=y is an equivalence relation.

For option (c)

(x,y)Sx-y is a multiple of 3

Reflexive: Let xS

x-x is always multiple of 3 then (x,x)S

Therefore, S is a reflexive relation.

Symmetric: Let x,yS

x-y is a multiple of 3

=-y-x is a multiple of 3

(y,x)S

Therefore, S is a symmetric relation.

Transitive: Let x,y,zS

(x,y)Sx-y is a multiple of 3 and (y,z)Sy-z is a multiple of 3

=x-y+y-z

=x-z is a multiple of 3

(x,z)S

Therefore, S is a transitive relation.

Hence, S is an equivalence relation.

For option (d)

(x,y)S if x-y is even

Reflexive: Let xS

x-x is always even then (x,x)S

Therefore, S is a reflexive relation.

Symmetric: Let x,yS

x-y

y-x is even

(y,x)S

Therefore, S is a symmetric relation.

Transitive: Let x,y,zS

(x,y)S if x-y is even and (y,z)S if y-z is even

x-y+y-z is even

=x-z is even

(x,z)S

Therefore, S is a transitive relation.

Hence, S is an equivalence relation.

Hence, the correct option is option (a).


flag
Suggest Corrections
thumbs-up
1
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Types of Relations
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon