Consider the following relation on set $B=\{a,b,\{a\},\{b\},\{a,b\}\}$ :

$P=\{(a,b),(b,\{a,b\}),(\{a,b\},a),(\{b\},a),(a,\{a\})\}.$

Which one of the following sets is a partition $S$ of $B=\{a,b,\{a\},\{b\},\{a,b\}\}?$

$($A partition of the given set $B$ can be defined as a set $S=\{S_1,S_2,S_3,$dots$\}.$ The members of $S$ are subsets of $B($each set $S_i$ is called a part of $S)$ such that

a$.$ for all $i,S_i($that is$,$ each part is nonempty$),$

b$.$ for all i and $j,$ if $S_i{S}_j,$ then $S_i{S}_j=($that is$,$ different parts have nothing in common$),$ and

c$.S_1{S}_2{S}_3$dots$=B($that is$,$ every element in $B$ is in some part $S_i).$

It is possible to form different partitions of $B$ depending on which subsets of $B$ are formed to be elements of $S.$

Test whether the sets given in the different alternatives meet all the criteria given in the above definition. Note that the elements of a partition of $B$ must be subsets of $B.$ Subsets of $B$ are formed when you keep the outside brackets of $B$ and then throw away all, some or no element of $B.$ For example, keep the outside brackets of $B,$ then throw away the element $\{a\},$ then the subset $\{a,b,\{b\},\{a,b\}\}$ of $B$ are formed. Refer to study guide, pp $94,95.)$

$$ a$.\{\{a,b,\{a\},\{b\}\},\{\{a,b\}\}\}$

b$.\{\{a\},\{b\},\{a,b\}\}$

c$.\{\{a,b,\{a\}\},\{\{a\},\{b\},\{a,b\}\}\}$

d$.\{a,b,\{a\},\{b\},\{a,b\}\}$