Educational

Check Function \(f(x)=x^2\) on \( \mathbb{Z} \) 📺 Video Explanation 📝 Question Check whether the function \[ f:\mathbb{Z}\to\mathbb{Z},\quad f(x)=x^2 \] is: injection (one-one) surjection (onto) bijection ✅ Solution 🔹 Step 1: Check Injection (One-One) A function is one-one if: \[ f(x_1)=f(x_2)\Rightarrow x_1=x_2 \] Take: \[ x_1=2,\quad x_2=-2 \] Then: \[ f(2)=4,\quad f(-2)=4 \] But: \[ […]

Check Function \(f(x)=x^2\) on \( \mathbb{N} \) 📺 Video Explanation 📝 Question Check whether the function \[ f:\mathbb{N}\to\mathbb{N},\quad f(x)=x^2 \] is: injection (one-one) surjection (onto) bijection ✅ Solution 🔹 Step 1: Check Injection (One-One) Assume: \[ f(x_1)=f(x_2) \] Then: \[ x_1^2=x_2^2 \] Since \(x_1,x_2\in\mathbb{N}\), both are positive. So: \[ x_1=x_2 \] ✔ Hence, function is

Show \(f(x)=x^2\) is Neither One-One Nor Onto 📺 Video Explanation 📝 Question Let: \[ A=\{-1,0,1\} \] and function: \[ f=\{(x,x^2):x\in A\} \] Show that: \[ f:A\to A \] is neither one-one nor onto. ✅ Solution 🔹 Step 1: Find Function Values For: \(f(-1)=(-1)^2=1\) \(f(0)=0^2=0\) \(f(1)=1^2=1\) So: \[ f=\{(-1,1),(0,0),(1,1)\} \] 🔹 Step 2: Check One-One A

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: \[ x_1=x_2 \]