Question

from the sets given below select equal and equivalent sets:

A={0,a} B={1,2,3,4} C={4,8,12} D={3,1,2,4} E={1,0} F={8,4,12} G={1,5,7,11} H={a,b}

Answer 1

Equivalent sets: If two sets have the same number of element then these sets are equivalent.
Equal sets: Two sets A and B are equal sets if every element of A is in B and
every element of B is in A.
$$\begin{gathered} A=\left\{0,a\right\}\Rightarrow n\left(A\right)=2 \\ B=\left\{1,2,3,4\right\}\Rightarrow n\left(B\right)=4 \\ C=\left\{4,8,12\right\}\Rightarrow n\left(C\right)=3 \\ D=\left\{3,1,2,4\right\}\Rightarrow n\left(D\right)=4 \\ E=\left\{1,0\right\}\Rightarrow n\left(E\right)=2 \\ F=\left\{8,4,12\right\}\Rightarrow n\left(F\right)=3 \\ G=\left\{1,5,7,11\right\}\Rightarrow n\left(G\right)=4 \\ H=\left\{a,b\right\}\Rightarrow n\left(H\right)=2 \\ Since,0\in Abut0doesnotbelongtoanysetexceptEbuta\notin E \\ Therefore,Aisnotequaltoanyset. \\ Similarly,E,G,andHarenotequaltoanyset. \\ And,,everyelementofBisinDandeveryelementofDisinB. \\ So,\mathit{B}\mathbf{=}\mathit{D} \\ And,everyelementofCisinFandeveryelementofFisinC. \\ So,\mathit{C}\mathbf{=}\mathit{F} \end{gathered}$$
$$\begin{gathered} Since,everyequalsetsareequivalentsets.Therefore,B\leftrightarrow D,C\leftrightarrow F \\ And,n\left(B\right)=n\left(G\right)\Rightarrow B\leftrightarrow G \\ n\left(A\right)=n\left(H\right)=n\left(E\right)\Rightarrow A\leftrightarrow H\leftrightarrow E \end{gathered}$$