New column with random boolean values while controlling the ratio of TRUE/FALSE per category

In R I've got a dataset like this one:

df <- data.frame(
  ID = c(1:30),
  x1 = seq(0, 1, length.out = 30),
  x2 = seq(100, 3000, length.out = 30),
  category = gl(3, 10, labels = c("NEGATIVE", "NEUTRAL", "POSITIVE"))
)

Now I want to add a new column with randomized boolean values, but inside each category the proportion of TRUE and FALSE values should be the same (i.e. the randomizing process should generate the same count of true and false values, in the above data frame 5 TRUEs and 5 FALSEs in each of the 3 categories). How to do this?