Ravi Kant Kumar

Prove \(f(x)=x^2+x+1\) is One-One but Not Onto 📺 Video Explanation 📝 Question Prove that the function \[ f:\mathbb{N}\to\mathbb{N}, \quad f(x)=x^2+x+1 \] is one-one but not onto. ✅ Solution 🔹 Step 1: Prove One-One (Injective) Assume: \[ f(x_1)=f(x_2) \] Then: \[ x_1^2+x_1+1=x_2^2+x_2+1 \] Cancel 1: \[ x_1^2+x_1=x_2^2+x_2 \] Rearrange: \[ x_1^2-x_2^2+x_1-x_2=0 \] Factor: \[ (x_1-x_2)(x_1+x_2+1)=0 \] […]

Check Whether Given Function is One-One and Onto 📺 Video Explanation 📝 Question Given: \[ A=\{a,b,c,d\},\quad B=\{x,y,z\} \] and function: \[ f=\{(a,x),(b,x),(c,z),(d,z)\} \] Check whether the function is one-one and onto. ✅ Solution 🔹 Check One-One (Injective) A function is one-one if different inputs have different outputs. Here: \(a \mapsto x\) \(b \mapsto x\) Different

Check Whether Given Function is One-One and Onto 📺 Video Explanation 📝 Question Given: \[ A=\{2,3,4\},\quad B=\{a,b,c\} \] and function: \[ f=\{(2,a),(3,b),(4,c)\} \] Check whether the function is one-one and onto. ✅ Solution 🔹 Check One-One (Injective) A function is one-one if different inputs have different outputs. Here: \(2 \mapsto a\) \(3 \mapsto b\) \(4

Check Whether Given Function is One-One and Onto 📺 Video Explanation 📝 Question Given: \[ A=\{1,2,3\},\quad B=\{3,5,7\} \] and function: \[ f=\{(1,3),(2,5),(3,7)\} \] Check whether the function is one-one and onto. ✅ Solution 🔹 Check One-One (Injective) A function is one-one if different inputs have different outputs. Here: \(1 \mapsto 3\) \(2 \mapsto 5\) \(3

Example of a Function Which is Neither One-One Nor Onto 📺 Video Explanation 📝 Question Give an example of a function which is: (iii) neither one-one nor onto. ✅ Solution Consider the function: \[ f:\mathbb{R}\to\mathbb{R} \] defined by: \[ f(x)=x^2 \] 🔹 Check One-One Take: \[ x=2,\quad x=-2 \] Then: \[ f(2)=4,\quad f(-2)=4 \] Different

Example of a Function Which is Onto but Not One-One 📺 Video Explanation 📝 Question Give an example of a function which is: (ii) not one-one but onto. ✅ Solution Consider the function: \[ f:\mathbb{R}\to [0,\infty) \] defined by: \[ f(x)=x^2 \] 🔹 Check One-One Take: \[ x=2,\quad x=-2 \] Then: \[ f(2)=4,\quad f(-2)=4 \]

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