Mathematics
Introduction to Relations
संबंधों का परिचय
In Class 12 Mathematics, under the chapter Relations and Functions, Introduction to Relations explains a relation as a subset of a Cartesian product. Students learn to form and count relations on sets, and identify important examples such as empty, universal and identity relations. The topic also introduces conditions used to recognise reflexive and symmetric relations.
Practice questions
On (A={1,2,3,4}), (R={(a,b):a+b\text{ is even}}) and (S={(a,b):a-b\text{ is divisible by }4}). How many pairs are in (R\cap S)?If (S={1,2,3}) and \(A=\mathcal{P}(S)\). On (A), (R={(X,Y):X\cap Y=X}). What is (R) with respect to reflexivity?For \(S={1,2,3}\) and \(A=\mathcal{P}(S)\), \(R={(X,Y):X\cup Y=Y}\). Is (R) reflexive?If \(S=\{1,2,3\}\) and \(A=\mathcal{P}(S)\), then a relation \(R\) on \(A\) is defined by \[R=\{(X,Y):X,Y\in A \text{ and } X\cup =S\}.
\] How many diagonal pairs must be added to make \(R\) reflexive?On \(A=\mathcal{P}({1,2,3})\), \(R={(X,Y):X\setminus Y=X}\). How many diagonal pairs are in (R)?If \(A=\mathcal{P}({1,2,3})) and (R={(X,Y):X\triangle Y\subseteq X}\), what is (R) with respect to reflexivity?If (f(x)=x+4), what is (f(x)-f(0))?If \(f(x)=2x+1\), what is the value of \(f(0)+f(1)\)?If \(f(x)=x^2+2\), what is \(f(2)-f(1)\)?If \(f(x)=\frac{x-2}{3}\), find the value of \(f(8)\).If \(f(x)=x-3\), what is \(f(10)-f(4)\)?If \(f(x)=x+5\) and \(g(x)=x^2\), what is \((f+g)(1)\)?If \(f(x)=x+1\) and \(g(x)=x+3\), what is \((fg)(1)\)?If \(f(x)=x+2\) and \(g(x)=3x\), what is \((f\circ g)(2)\)?If \(f(x)=x+1\), \(g(x)=2x\), and \(h(x)=x^2\), what is the value of \((h\circ g\circ f)(1)\)?If \(f(x)=x-2\) and \(g(x)=x^2+1\), find the value of \((g\circ f)(4)\).If (f:R\to R), (f(x)=4x+7), what is the value of (f^{-1}(23))?If (f(x)=x^2-4x), what is (f(x+1)-f(x))?If (f(x)=\frac{x-5}{x+4}), what is the real domain?If (f:R\to R), (f(x)=x^2-6x+11), what is the range of (f)?