True or False
The ordered air (5,2) belongs to the relations R = {(x,y): y = x-5,x,y`in`Z}.

Let N be the set of natural numbers and the relation R be defined on N such that R = {(x, y) : y = 2x, x, y in N} . Is this relation a function ?

True or false The line x=y=z lies in the plane x-2y+z-1=0.

Prove that the following relation R in Z of integers is an equivalence relation R = {(x, y): x-y is an integer}

Prove that the following relation R in Z of integers is an equivalence relation : R = [ (x, y) : x - y is an integer}

Prove that the following relation R in Z of integers is an equivalence relation : R = [ (x, y) : 2x - 2y is an integer}

Prove that the following relation R in Z of integers is an equivalence relation : R = [ (x, y) : 3x- 3y is an integer}

Let X = {1, 2, 3, 4, 5, 6, 7, 8, 9} . Let R_1 be a relation in X given by R_1 = {(x, y) : x – y is divisible by 3} and R_2 be another relation on X given by R_2 = {(x, y): {x, y} sub {1, 4, 7} or {x, y} sub {2, 5, 8} or {(x, y} sub {3, 6, 9} . Show that R_1 = R_2 .

Determine whether each of the following relations are reflexive, symmetric and transitive : (i) Relation R in the set A = {1, 2, 3,......, 13, 14} defined as R={(x,y): 3x-y=0} (ii) Relation R in the set N of natural numbers defined as R={(x,y): y=x+5 " and " x lt 4} (iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} defined as R = {(x, y): y is divisible by x}. iv) Relation R in the set Z, of all integers defined as R = {(x, y) : x -y is an integer}.

Show that relation R in Z of integers given by R = {x,y} : x - y is divisible by 5, x , y inZ } is an equivalence relation.

Using matrices, solve the following system of equations for x,y and z: 2x+3y + 3z=5, x-2y + z = -4, 3x-y - 2z=3