Chpt. 1 Sets, Relations, and Arguments Flashcards
A Set
A set is a collection of objects.
An Element
An element of a set is an object within the collection.
Identical Sets
Identical sets are sets with the same elements.
What does “a ∈ A” mean?
That a is an element of set A.
What does ∅ represent?
An empty set that contains no elements.
How do read this: {x: x is an animal with a heart}
The set of all animals with a heart.
Write this in set notation: the set of all UN recognized states.
{x: x is a UN recognized state}
A Binary Relation
A binary relation is a set that contains only ordered pairs.
Is ∅ a binary relation?
Yes, because it does not contain anything that is not an ordered pair.
A binary relation R is reflexive on a set S if and only if…
for all elements d of S the pair [d,d] is an element of R
A binary relation R is symmetric on a set S if and only if…
for all elements d, e of S the pairs [d,e] ∈ R and [e,d] ∈ R
A binary relation R is asymmetric on a set S if and only if…
for no elements d, e of S the pair [d,e] ∈ R and [e,d] ∈ R
A binary relation R is antisymmetric on a set S if and only if…
for no two distinct elements of d, e of S, the pairs [d,e] ∈ R and [e,d] ∈ R
A binary relation R is transitive on a set S if and only if…
for all elements d, e, f of S: if [d,e] ∈ R and [e,f] ∈ R then also [d,f] ∈ R
A binary relation R is symmetric if and only if…
it is symmetric on all sets; for all d,e: if [d,e] ∈ R then [e,d] ∈ R
