April 2026

Example of a One-One but Not Onto Function 📺 Video Explanation 📝 Question Give an example of a function which is: (i) one-one but not onto. ✅ Solution Consider the function: \[ f:\mathbb{N}\to\mathbb{N} \] defined by: \[ f(x)=x+1 \] 🔹 Check One-One (Injective) Assume: \[ f(x_1)=f(x_2) \] Then: \[ x_1+1=x_2+1 \] So: \[ x_1=x_2 \] […]

Relation \( x-y+\sqrt{2} \) Irrational on Real Numbers 📺 Video Explanation 📝 Question For real numbers \(x\) and \(y\), relation \(R\) is defined by: \[ xRy \iff x-y+\sqrt{2} \text{ is irrational} \] Then, \(R\) is: A. reflexive B. symmetric C. transitive D. none of these ✅ Solution 🔹 Reflexive Check Put: \[ x=y \] Then:

Brother Relation in a Family 📺 Video Explanation 📝 Question Let a non-empty set consist of children in a family. A relation \(R\) is defined by: \[ aRb \iff a \text{ is brother of } b \] Then, \(R\) is: A. symmetric but not transitive B. transitive but not symmetric C. neither symmetric nor transitive

Congruence Relation on Set of Triangles 📺 Video Explanation 📝 Question Let \(T\) be the set of all triangles in the Euclidean plane. A relation \(R\) on \(T\) is defined by: \[ aRb \iff a \text{ is congruent to } b \] Then, \(R\) is: A. reflexive but not symmetric B. transitive but not symmetric

Relation of Perpendicular Lines in a Plane 📺 Video Explanation 📝 Question Let \(L\) denote the set of all straight lines in a plane. A relation \(R\) is defined by: \[ lRm \iff l \perp m \] Then, \(R\) is: A. reflexive B. symmetric C. transitive D. none of these ✅ Solution 🔹 Reflexive Check

Divisibility Relation on \( \mathbb{N} \) 📺 Video Explanation 📝 Question Let relation \(R\) on natural numbers \( \mathbb{N} \) be defined by: \[ nRm \iff n \text{ divides } m \] Then, \(R\) is: A. Reflexive and symmetric B. Transitive and symmetric C. Equivalence relation D. Reflexive, transitive but not symmetric ✅ Solution 🔹

Maximum Number of Equivalence Relations on Set \( A=\{1,2,3\} \) 📺 Video Explanation 📝 Question Find the maximum number of equivalence relations on the set: \[ A=\{1,2,3\} \] A. 1 B. 2 C. 3 D. 5 ✅ Solution Number of equivalence relations on a finite set = number of partitions of that set. For set

Relation \( a\geq b \) on \( \mathbb{R} \) 📺 Video Explanation 📝 Question Let relation \(S\) on the set of real numbers \(\mathbb{R}\) be defined by: \[ aSb \iff a\geq b \] Then, \(S\) is: A. an equivalence relation B. reflexive, transitive but not symmetric C. symmetric, transitive but not reflexive D. neither transitive

Relation on Set \( A=\{1,2,3\} \) 📺 Video Explanation 📝 Question Let: \[ A=\{1,2,3\} \] and: \[ R=\{(1,1),(2,2),(3,3),(1,2),(2,3),(1,3)\} \] Then, \(R\) is: A. reflexive but not symmetric B. reflexive but not transitive C. symmetric and transitive D. neither symmetric nor transitive ✅ Solution 🔹 Reflexive Check A relation is reflexive if: \[ (1,1),(2,2),(3,3) \] are

Which Relation on \( \mathbb{Z} \) is Not an Equivalence Relation? 📺 Video Explanation 📝 Question Let \( \mathbb{Z} \) be the set of all integers. Which of the following relations is not an equivalence relation? A. \(xRy \iff x\leq y\) B. \(xRy \iff x=y\) C. \(xRy \iff x-y \text{ is even}\) D. \(xRy \iff