I found that the package VennDiagram has a function calculate.overlap() but I wasn't able to find a way to name the sections from this function. However, if you use package gplots , there is the function venn() which will return the intersections attribute.

AW.DL <- c("a","b","c","d")

AW.FL <- c("a","b", "e", "f")

AW.UL <- c("a","c", "e", "g")

library(gplots)

lst <- list(AW.DL,AW.FL,AW.UL)

ItemsList <- venn(lst, show.plot = FALSE)

lengths(attributes(ItemsList)$intersections)

Output:

> lengths(attributes(ItemsList)$intersections)
    A     B     C   A:B   A:C   B:C A:B:C 
    1     1     1     1     1     1     1

To get elements, just print attributes(ItemsList)$intersections:

> attributes(ItemsList)$intersections
$A
[1] "d"

$B
[1] "f"

$C
[1] "g"

$`A:B`
[1] "b"

$`A:C`
[1] "c"

$`B:C`
[1] "e"

$`A:B:C`
[1] "a"

Take a look at ?intersect, ?union and ?setdiff function to extract the different fields of the Venn diagram.

I have created some list versions of the two functions to better get the elements in the different compartments:

Intersect <- function (x) {  
  # Multiple set version of intersect
  # x is a list
  if (length(x) == 1) {
    unlist(x)
  } else if (length(x) == 2) {
    intersect(x[[1]], x[[2]])
  } else if (length(x) > 2){
    intersect(x[[1]], Intersect(x[-1]))
  }
}

Union <- function (x) {  
  # Multiple set version of union
  # x is a list
  if (length(x) == 1) {
    unlist(x)
  } else if (length(x) == 2) {
    union(x[[1]], x[[2]])
  } else if (length(x) > 2) {
    union(x[[1]], Union(x[-1]))
  }
}

Setdiff <- function (x, y) {
  # Remove the union of the y's from the common x's. 
  # x and y are lists of characters.
  xx <- Intersect(x)
  yy <- Union(y)
  setdiff(xx, yy)
}

So, if we want to see the common elements (i.e. the union of A, B, C, and D) or the ones in C and D but not in A and B in your example we do something like the following.

set.seed(1)
xx.1 <- list(A = sample(LETTERS, 15), 
             B = sample(LETTERS, 15), 
             C = sample(LETTERS, 15), 
             D = sample(LETTERS, 15))
Intersect(xx.1)
#[1] "E" "L"
Setdiff(xx.1[c("C", "D")], xx.1[c("A", "B")])
#[1] "O" "P" "K" "H"

Hope this helps!

Edit: Systematically get all components

By some (I think) clever use of the combn function, indexing, and a good understanding of lapply we can all elements systematically:

# Create a list of all the combinations
combs <- 
  unlist(lapply(1:length(xx.1), 
                function(j) combn(names(xx.1), j, simplify = FALSE)),
         recursive = FALSE)
names(combs) <- sapply(combs, function(i) paste0(i, collapse = ""))
str(combs)
#List of 15
# $ A   : chr "A"
# $ B   : chr "B"
# $ C   : chr "C"
# $ D   : chr "D"
# $ AB  : chr [1:2] "A" "B"
# $ AC  : chr [1:2] "A" "C"
# $ AD  : chr [1:2] "A" "D"
# $ BC  : chr [1:2] "B" "C"
# $ BD  : chr [1:2] "B" "D"
# $ CD  : chr [1:2] "C" "D"
# $ ABC : chr [1:3] "A" "B" "C"
# $ ABD : chr [1:3] "A" "B" "D"
# $ ACD : chr [1:3] "A" "C" "D"
# $ BCD : chr [1:3] "B" "C" "D"
# $ ABCD: chr [1:4] "A" "B" "C" "D"

# "A" means "everything in A minus all others"
# "A", "B" means "everything in "A" and "B" minus all others" and so on
elements <- 
  lapply(combs, function(i) Setdiff(xx.1[i], xx.1[setdiff(names(xx.1), i)]))

n.elements <- sapply(elements, length)
print(n.elements)
#   A    B    C    D   AB   AC   AD   BC   BD   CD  ABC  ABD  ACD  BCD ABCD 
#   2    2    0    0    1    2    2    0    3    4    4    1    1    2    2 

Another simple example that shows how to print a Venn diagram using the VennDiagram package:

library(VennDiagram)
cardiome <- letters[1:10]
superset <- letters[8:24]
overlap <- calculate.overlap(
x <- list("Cardiome"=cardiome, "SuperSet"=superset))

venn.plot <- draw.pairwise.venn(
    area1 = length(cardiome),
    area2 = length(superset),
    cross.area = length(overlap),
    category = c("Cardiome", "Superset"),
    fill = c("blue", "red"),
    lty = "blank",
    cex = 2,
    cat.cex = 2,
    cat.pos = c(180, 180),
    cat.dist = 0.05,
    cat.just = list(c(0, 1), c(1, 1))
    )
grid.draw(venn.plot)
savePlot(filename="venndiag", type="png")

Venn diagrams with item labels inside the sets:

library(RAM)
vectors <- list(Cardiome=cardiome, Superset=superset)
group.venn(vectors=vectors, label=TRUE, 
    fill = c("blue", "red"),
    cat.pos = c(180, 180),
    lab.cex=1.1)

By replacing x in overlap[[x]] with red number 1-15, you can get a complete list of genes of interest at the specific location in Venn diagram.

Also, you can get the numbers of genes by using length() function.