Given the set $A = \{a,b,c,d\}$
The number of subsets is $2^4 = 16$
The number of subsets with $3$ elements is ${4 \choose 3} = 4$
The number of functions from $A$ to $A$ is $4^4=256$
The number of surjective functions from $A$ to $A$ is $$ \sum_{i=0}^{3} (-1)^i{4 \choose i}(4-i)^4 = 24 $$
The number of binary relations is $2^{(4^{2})} = 65536$
The number of equivalence relations is $15$ (numbers of Bell)
Please tell me if it's all right or if I made some mistakes, thanks.
All your answers are correct.
The number of subsets is $2^4=16$ is a special case of $|P(A)|=2^{|A|}$
The number of functions from $A$ to $A$ is a special case of the number of functions from $A$ to $B$ equals $|B|^{|A|}.$
The number of surjective functions from $A$ to $A$ in this case is simply $4!=24$ because you have a set of 4 elements and each surjective function is necessarily a permutation on elements of $A$
The number of binary relations is $ 2^{(4^2)}=65536$ because binary relations are subsets of $A\times A$ and you have $|A|=4$ thus $|A\times A|=16= 4^{2}.$
The number of equivalence relations is $15$ (numbers of Bell) because the number of equivalence relations are the same as the number of partitions on $A$