Domain of Relation \( a^2 + b^2 = 25 \) 📺 Video Explanation 📝 Question Let relation \( R \) on \( \mathbb{Z} \) be defined as: \[ (a,b) \in R \iff a^2 + b^2 = 25 \] Find the domain of \( R \). ✅ Solution 🔹 Step 1: Understand Domain Domain = set […]
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: \[ A = \{0,1,2\} \]
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) \(
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
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
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
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
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
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