Check if one element dominates the other.

## Usage

dominates(powerRelation, e1, e2, strictly = FALSE, includeEmptySet = TRUE)

## Arguments

powerRelation

A PowerRelation object created by PowerRelation() or as.PowerRelation()

e1, e2

Elements in powerRelation\$elements

strictly

If TRUE, check if p1 strictly dominates p2

includeEmptySet

If TRUE, check $$\lbrace i \rbrace \succsim \lbrace j \rbrace$$ even if empty set is not part of the power relation.

## Value

Logical value TRUE if e1 dominates e2, else FALSE.

## Details

$$i$$ is said to dominate $$j$$ if $$S \cup \lbrace i \rbrace \succsim S \cup \lbrace j \rbrace$$ for all $$S \in 2^{N \setminus \lbrace i,j \rbrace}$$.

$$i$$ strictly dominates $$j$$ if there also exists an $$S \in 2^{N \setminus \lbrace i,j \rbrace}$$ such that $$S \cup \lbrace i \rbrace \succ S \cup \lbrace j \rbrace$$.

## Examples

pr <- as.PowerRelation("12 > 1 > 2")

# TRUE
d1 <- dominates(pr, 1, 2)

# FALSE
d2 <- dominates(pr, 2, 1)

# TRUE (because it's not strict dominance)
d3 <- dominates(pr, 1, 1)

# FALSE
d4 <- dominates(pr, 1, 1, strictly = TRUE)

stopifnot(all(d1, !d2, d3, !d4))