a <- "Roy lives in Japan and travels to Africa"
b <- "Roy travels Africa with this wife"

stra <- as.data.frame(t(read.table(textConnection(a), sep = " ")))
strb <- as.data.frame(t(read.table(textConnection(b), sep = " ")))

stra_unique <-as.data.frame(unique(stra$V1))
strb_unique <- as.data.frame(unique(strb$V1))
colnames(stra_unique) <- c("V1")
colnames(strb_unique) <- c("V1")

common_words <-length(merge(stra_unique,strb_unique, by = "V1")$V1)

> a <- "Roy lives in Japan and travels to Africa"
> b <- "Roy travels Africa with this wife"
> a_split <- unlist(strsplit(a, sep=" "))
> b_split <- unlist(strsplit(b, sep=" "))
> length(intersect(a_split, b_split))
[1] 3

vec1 <- c(a,b)
 Reduce(`intersect`,str_extract_all(vec1, "\w+"))
 #[1] "Roy"     "travels" "Africa"

library(stringi)
 Reduce(`intersect`,stri_extract_all_regex(vec1,"\w+"))
 #[1] "Roy"     "travels" "Africa"

length(Reduce(`intersect`,stri_extract_all_regex(vec1,"\w+")))
 #[1] 3

Reduce(`intersect`,regmatches(vec1,gregexpr("\w+", vec1)))
  #[1] "Roy"     "travels" "Africa"

a <- "Roy lives in Japan and travels to Africa"
b <- "Roy travels Africa with this wife"
c <- "Bob also travels Africa for trips but lives in the US unlike Roy."

library(stringi);library(qdapTools)
X <- stri_extract_all_words(list(a, b, c))
X <- mtabulate(X) > 0
Y <- colSums(X) == nrow(X); names(Y)[Y]

[1] "Africa"  "Roy"     "travels"