This is clearly a problem on sets. You just need to count over the intersection of the two sets. Try these functions:

sets {base}

Simple example:

a <- c(1:10)
b <- c(5:15)
a
[1]  1  2  3  4  5  6  7  8  9 10
b
[1]  5  6  7  8  9 10 11 12 13 14 15
intersect(a,b)
[1]  5  6  7  8  9 10
#Looping part
for(i in 1:10){
    c <- sample(a, 5)
    intersect(c, b)
}

Now, it's up to you adapt the code. I suggest you to read about apply functions. They are quite handy in situations like yours. If your elements are intervals of coordinates you could use findInterval.

It is helpful if you give us context, how is this related to bioinformatics, what are these numbers? Otherwise, it is just another programming question more suitable for stackoverflow.

Here is my try:

#dummy data
# a is a bucket of numbers
a <- c(1:100)
# b is a small selection of numbers
b <- round(runif(10,1,100))

# do 10 times: if we pick randomly 10 numbers from bucket
#              how many overlap with the b
output <- sapply(1:10,
                 function(TrialCount)
                   c(TrialCount,length(intersect(sample(a,10), b))))

# pretty output
output <- as.data.frame(t(output))
colnames(output) <- c("Trial(n)","Intersections(y)")

I think that's a fair solution for solving the problem related to overlapping elements, but there's a critical gap remaining. I intend to perform this "intersection" on only a sample of a items, yet need to do so over and over. I'm not concerned with the knowing the absolute intersection, only the intersection from smaller samples of a to the whole b.
How might I perform this iteratively with a simple loop, and ask to write the ouput into a text file formatted as follows:

Trial (n) Intersections (y)

[1] 500

[2] 154

[3] 10

[4] 95

.

.

.

[10000] 299