Let $A=\{a,b,c,d\},$ define an equivalence relation $R$ on $A,$

$R=\{($:$a,b$:$),($:$b,a$:$),($:$c,d$:$),($:$d,c$:$)\}{I}_A$

Draw the relation diagram of $R,$ and find equivalence classes of each element in $A.$

$2.$ For any non$-$empty set A and being a non$-$empty family of sets in $A,$ does constitute a partition of $A?$

$3.$ Let $A=\{1,2,3,4\},$ define a binary relation $R$ on $AA,$

$AA($:$u,v$:$),($:$x,y$:$)inAA,($:$u,v$:$)R($:$x,y$:$)>u+y=x+v$

$(1)$ Prove that $R$ is an equivalence relation on $AA.$

$(2)$ Determine the partition of $A$A caused by $R.$

$4.$ Draw the Hasse diagram for the following sets with the divisibility relation.

$\{1,2,3,4,5,6,7,8,9,10,11,12\}.$

Draw the Hasse diagram of the following poset