Let's set vectors

Group1<-c("A","R","t","Y","p","Q")

Group2<-c("Z","R","t","O","p","X")

> Group1

[1] "A" "R" "t" "Y" "p" "Q"

> Group2

[1] "Z" "R" "t" "O" "p" "X"

Intersection: there are 2 ways

> intersect(Group1,Group2)

[1] "R" "t" "p"

> Group1[which(Group1%in%Group2)]

[1] "R" "t" "p"

Union:

> union(Group1,Group2)

[1] "A" "R" "t" "Y" "p" "Q" "Z" "O" "X"

Elements exclusive to Group 1: there are 2 ways

> setdiff(Group1,Group2)

[1] "A" "Y" "Q"

> Group1[which(!Group1%in%Group2)]

[1] "A" "Y" "Q"

Elements exclusive to Group 2: there are 2 ways

> setdiff(Group2,Group1)

[1] "Z" "O" "X"

> Group2[which(!Group2%in%Group1)]

[1] "Z" "O" "X"

Mutually exclusive elements

> union(setdiff(Group1,Group2),setdiff(Group2,Group1))

[1] "A" "Y" "Q" "Z" "O" "X"

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