If R and S are transitive relations on a set A, then prove that R ⋃ S may not be a transitive relation on A.
Union of Two Transitive Relations 📺 Video Explanation 📝 Question If \( R \) and \( S \) are transitive relations on a set \( A \), show that: \[ R \cup S \text{ may not be transitive} \] ✅ Solution (By Counterexample) 🔹 Step 1: Take a Set Let: \[ A = \{0,1,2\} \]
If R and S are relations on a set A, then prove the following (i)R and S are symmetric ⇔ R ⋂ S, and R ⋃ S is symmetric (ii) R is reflexive, and S is any relation ⇔ R ⋃ S is reflexive.
Union and Intersection of Relations 📺 Video Explanation 📝 Question If \( R \) and \( S \) are relations on a set \( A \), prove: (i) \( R \) and \( S \) are symmetric \( \iff \) \( R \cap S \) and \( R \cup S \) are symmetric (ii) \(
Let Z be the set of all integers and Z0 be the set of all non-zero integers. Let a relation R on Z × Z0 be defined as follows : (a, b) R (c, d) ⇔ ad = bc for all (a, b), (c, d) ∈ Z × Z0 Prove that R is an equivalence relation on Z × Z0
Relation \( ad = bc \) on \( \mathbb{Z} \times \mathbb{Z}_0 \) 📺 Video Explanation 📝 Question Let \( \mathbb{Z} \) be the set of integers and \( \mathbb{Z}_0 \) the set of non-zero integers. Define relation \( R \) on \( \mathbb{Z} \times \mathbb{Z}_0 \) as: \[ (a,b) R (c,d) \iff ad = bc
Let S be a relation on the set R of all real numbers defined by S = {(a, b) ∈ R × R : a^2 + b^2 = 1}. Prove that S is not an equivalence relation on R.
Relation \( a^2 + b^2 = 1 \) on \( \mathbb{R} \) 📺 Video Explanation 📝 Question Let relation \( S \) on \( \mathbb{R} \) be defined as: \[ (a, b) \in S \iff a^2 + b^2 = 1 \] Show that \( S \) is not an equivalence relation. ✅ Concept Used A
Let R be the relation defined on the set A = {1, 2, 3, 4, 5, 6, 7} by R = {(a, b) : both a and b are either odd or even}. Show that R is an equivalence relation. Further, show that all the elements of the subset {1, 3, 5, 7} are related to each other, and all the elements of the subset {2, 4, 6} are related to each other, but no element of the subset {1, 3, 5, 7} is related to any element of the subset {2, 4, 6}.
Relation Based on Even and Odd Numbers 📺 Video Explanation 📝 Question Let \( A = \{1,2,3,4,5,6,7\} \). Define relation \( R \) as: \[ (a,b) \in R \iff \text{both } a \text{ and } b \text{ are either odd or even} \] Show that \( R \) is an equivalence relation. Also analyze the
Let O be the origin. We define a relation between two points P and Q in a plane if OP = OQ. Show that the relation, so defined is an equivalence relation.
Relation \( OP = OQ \) on Points in a Plane 📺 Video Explanation 📝 Question Let \( O \) be the origin. Define a relation \( R \) between two points \( P \) and \( Q \) in a plane as: \[ (P, Q) \in R \iff OP = OQ \] Show that
Show that the relation R, defined on the set A of all polygons as R = {(P1, P2) : P1 and P2 have same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right angle triangle T with sides 3, 4 and 5 ?
Relation on Polygons Having Same Number of Sides 📺 Video Explanation 📝 Question Let \( A \) be the set of all polygons. Define relation \( R \) as: \[ R = \{(P_1, P_2) : P_1 \text{ and } P_2 \text{ have same number of sides}\} \] Show that \( R \) is an equivalence
Let L be the set of all lines in XY-plane and R be the relation in L defined as R = {(L1, L2): L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.
Relation of Parallel Lines on XY-Plane 📺 Video Explanation 📝 Question Let \( L \) be the set of all lines in the XY-plane. Define relation \( R \) as: \[ R = \{(L_1, L_2) : L_1 \text{ is parallel to } L_2\} \] Show that \( R \) is an equivalence relation. Also find
Show that the relation R on the set A = {x ∈ Z ; 0 ≤ x ≤ 12}, given by R = {(a, b) : a = b}, is an equivalence relation. Find the set of all elements related to 1.
Relation \( a = b \) on Set \( A \) 📺 Video Explanation 📝 Question Let \( A = \{x \in \mathbb{Z} : 0 \le x \le 12\} \). Define relation \( R \) as: \[ R = \{(a,b) : a = b\} \] Show that \( R \) is an equivalence relation. Also