Let N be the set of all natural numbers and R be the relation in `NxxN` defined by (a,b) R (c,d), if ad=bc. Show that R is an equivalence relation.

Let L be the set of all lines in the XY - plane and R be the relation in L defined by R={(l_(i)l_(j))|l_(i)" is parallel to "l_(j)l_(i)AA_(i),j} . Show that R is an equivalence relation. Find the set of all lines related to the line y=7x+5

Let L be the set of all lines in a plane and R be the relation in L defined as R={(L_(1),L_(2)):L_(1)" is perpemdicular to "L_(2)} Show that R is symmetric but neither reflexive nor transitive.

Let L be the set of all straight lines in the XY plane and let R be a relation defined in L as R={(L_1,L_2) : L_1, L_2 in L} L_1 is parallel to L_2 Show that R is an equivalence relation. What is the set of lines related to the line y=2x+3 ?

Let A=NxxN and let ** be a binary operation on A defined by (a,b)**(c,d)=(a+c,b+d) . Show that ** is commutative and associative.

A binary relation R in the set A={2,3,4,5,6} is defined by xRy, where |x-y| is divisible by 3. Is R an equivalence relation?

If R={(1,2),(2,3)} is a relation in N, list the least number of ordered pairs to be included in R so that R is an equivalence relation.

Let T bt the set of all triangle in the coordinate plane. Show that the relations in T defined as R_1 = {(T_1,T_2) : T_1, T_2 in T} T_1 is similar to T_2 and R_2 = {(T_1, T_2) : T_1, T_2 in T} , T_1 is congruent to T_2 , are equivalence relations.

Show that the relation R in R, defined by R={(a,b),aleb} , is reflexive and transsitive but not symmetric.