Question - 11 : - Show that the relation R in the set A of punts in aplane given by R = {(P, Q) : distance of the point P from the origin is same asthe distance of the punt Q from the origin}, is an equivalence relation.Further, show that the set of all punts related to a point P ≠ (0,0) is thecircle passing through P with origin as centre.

Answer - 11 : -

Let O be the origin then the relation
R={(P,Q):OP=OQ}
(i) R is reflexive. Take any distance OP,
OP = OP => R is reflexive.
(ii) R is symmetric, if OP = OQ then OQ = OP
(iii) R is transitive, let OP = OQ and OQ = OR =>OP=OR
Hence, R is an equivalence relation.
Since OP = K (constant) => P lies on a circle with centre at the origin.

Question - 12 : - Show that the relation R defined in the set A of alltriangles as R= {(T1, T2): T1 is similar to T2}, is equivalence relation.Consider three right angle triangles T1 with sides 3,4,5, T2 with sides 5,12,13and T3 with sides 6,8,10. Which triangles among T1, T2 and T3 are related?

Answer - 12 : -

(i) In a set of triangles R = {(T1, T2) : T1 is similar T2}
(a) Since A triangle T is similar to itself. Therefore (T, T) ∈ R for all T ∈ A.
∴Since R is reflexive
(b) If triangle T1 is similar to triangle T2 then T2 is similar triangle T1
∴R is symmetric.
(c) Let T1 is similar to triangle T2 and T2 to T3 then triangle T1 is similarto triangle T3,
∴R is transitive.
Hence, R is an equivalence relation.
(ii) Two triangles are similar if their sides are proportional now sides 3,4,5of triangle T1 are proportional to the sides 6, 8, 10 of triangle T3.
∴T1 is related to T3.

Question - 13 : - Show that the relation R defined in the set A of allpolygons as R = {(P1, P2) : P1 and P2 have same number of sides}, is anequivalence relation. What is the set of all elements in A related to the rightangle triangle T with sides 3,4 and 5?

Answer - 13 : -

Let n be the number of sides of polygon P1.
R= {(P1, P2): P1 and P2 are n sides polygons}
(i) (a) Any polygon P1 has n sides => R is reflexive
(b) If P1 has n sides, P2 also has n sides then if P2 has n sides P1 also has nsides.
=> R is symmetric.
(c) Let P1, P2; P2, P3 are n sided polygons. P1 and P3 are also n sidedpolygons.
=> R is transitive. Hence R is an equivalence relation.
(ii) The set A = set of all the triangles in a plane.

Question - 14 : - Let L be the set of all lines in XY plane and R bethe relation in L defined as R={(L1, L2): L1 is parallel to L2}. Show that R isan equivalence relation. Find the set of all lines related to the line y =2x+4.

Answer - 14 : -

L = set of all the lines in XY plane, R= {(L1,L2) : L1 is parallelto L2}
(i) (a) L1 is parallel to itself => R is reflexive.
(b) L1 is parallel to L2 => L2 is parallel to L1 R is symmetric.
(c) Let L1 is parallel to L2 and L2 is parallel to L3 and L1 is parallel to L3=> R is transitive.
Hence, R is an equivalence relation.
(ii) Set of parallel lines related to y = 2x + 4 is y = 2x + c, where c is anarbitrary constant.

Question - 15 : - Let R be the relation in the set {1,2,3,4} given byR={(1,2), (2,2), (1,1), (4,4), (1,3), (3,3), (3,2) }. Choose the correctanswer.

Answer - 15 : - (a) R is reflexive and symmetric but not transitive.
(b) R is reflexive and transitive but not symmetric.
(c) R is symmetric and transitive but not reflexive.
(d) R is an equivalence relation.

Solution
The correct answer is B.

R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}

It is seen that (a, a) ∈ R, for every a ∈{1, 2, 3, 4}.

∴ R is reflexive.

It is seen that (1, 2) ∈ R, but (2, 1) ∉ R.

∴R is not symmetric.

Also, it is observed that (a, b), (b, c) ∈ R ⇒ (a, c) ∈ R for all a, b, c ∈ {1, 2, 3, 4}.

∴ R is transitive.

Hence, R is reflexive and transitive but not symmetric.

Question - 16 : - Let R be the relation in the set N given by R = {(a,b): a=b – 2, b > 6}. Choose the correct answer.

Answer - 16 : - (a)(2,4)∈R
(b)(3,8)∈R
(c)(6,8)∈R
(d)(8,7)∈R

Solution

Option (c) satisfies the condition that a = b – 2
i. e. 6 = 8 – 2 and b > 6, i.e. b = 8
=> option (c) is correct.

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