Relation \( x + y = 10 \) on \( \mathbb{N} \) 📺 Video Explanation 📝 Question Let relation \( R \) on \( \mathbb{N} \) be defined as: \[ (x, y) \in R \iff x + y = 10 \] Check whether \( R \) is reflexive, symmetric, and transitive. ✅ Solution 🔹 Step […]
Relation \( x > y \) on \( \mathbb{N} \) 📺 Video Explanation 📝 Question Let relation \( R \) on \( \mathbb{N} \) be defined as: \[ (x, y) \in R \iff x > y \] Check whether \( R \) is reflexive, symmetric, and transitive. ✅ Solution 🔹 Step 1: Reflexive A relation
Making a Relation Reflexive and Transitive 📺 Video Explanation 📝 Question Let \( A = \{a,b,c\} \) and \[ R = \{(a,a),(b,c),(a,b)\} \] Find the minimum number of ordered pairs to be added so that \( R \) becomes reflexive and transitive. ✅ Solution 🔹 Step 1: Make Reflexive Reflexive requires: \[ (a,a),(b,b),(c,c) \] Already
Making a Relation Transitive 📺 Video Explanation 📝 Question Let \( A = \{1,2,3\} \) and \[ R = \{(1,2),(1,1),(2,3)\} \] Find the minimum number of ordered pairs to be added so that \( R \) becomes transitive. ✅ Solution 🔹 Step 1: Check Transitivity Condition A relation is transitive if: \[ (a,b),(b,c) \in R
Making a Relation Reflexive, Symmetric and Transitive 📺 Video Explanation 📝 Question Given relation: \[ R = \{(1,2),(2,3)\} \text{ on } A = \{1,2,3\} \] Add minimum number of ordered pairs so that the relation becomes reflexive, symmetric and transitive. ✅ Solution 🔹 Step 1: Make Reflexive Reflexive requires: \[ (1,1),(2,2),(3,3) \] Add: \[ (1,1),(2,2),(3,3)
Examples of Relations with Given Properties 📺 Video Explanation 📝 Question Give examples of relations having the following properties: (i) Reflexive and symmetric but not transitive (ii) Reflexive and transitive but not symmetric (iii) Symmetric and transitive but not reflexive (iv) Symmetric but neither reflexive nor transitive (v) Transitive but neither reflexive nor symmetric ✅
Relation \( \geq \) on Real Numbers 📺 Video Explanation 📝 Question Show that the relation \( R \) defined on \( \mathbb{R} \) by: \[ (a, b) \in R \iff a \geq b \] is reflexive and transitive but not symmetric. ✅ Solution 🔹 Step 1: Reflexive A relation is reflexive if: \[ (a,
Relation: m is a Multiple of n 📺 Video Explanation 📝 Question A relation \( R \) on integers is defined as: \[ (m, n) \in R \iff m \text{ is a multiple of } n \] Check whether \( R \) is reflexive, symmetric, and transitive. ✅ Solution 🔹 Step 1: Understanding the Relation
Is Every Symmetric and Transitive Relation Reflexive? 📺 Video Explanation 📝 Question Is it true that every relation which is symmetric and transitive is also reflexive? Give reasons. ✅ Solution 🔹 Statement The statement is False. 🔹 Counterexample Let \( A = \{1,2\} \) Define relation: \[ R = \{(1,1)\} \] 🔹 Check Properties Symmetric:
Relation Defined by \( 2x + y = 41 \) on \( \mathbb{N} \) 📺 Video Explanation 📝 Question Let \( R = \{(x,y) : x,y \in \mathbb{N},\ 2x + y = 41\} \). Find: Domain of \( R \) Range of \( R \) Check whether \( R \) is reflexive, symmetric, and transitive