Binary Operators

Question 1

What does it mean when a property is closed?

Answer 1

An operator * is closed in a set S if for every pair of

elements a and b in S the quantity a*b ∈ S

Question 2

What does it mean when a property is Commutative?

Answer 2

An operator * is commutative in a set S if for every
pair of elements a and b in S
ab = ba

Question 3

What does it mean when a property is Associative?

Answer 3

An operator * is associative in a set S if for every three
elements a, b and c in S
(ab)c = a(bc)

Question 4

What does it mean when a property is Distributive?

Answer 4

An operator * distributes over an operator # in the set

S if for every three elements a, b and c in S a*(b # c) = (a * b) # (a * c)

Question 5

What does it mean when a property has an Identity?

Answer 5

An operator * has an identity in a set S if there is an

element e in S such that for every element a in S ae = ea = a

Question 6

What does it mean when a property has an Inverse?

Answer 6

An element a in S has an inverse a-1 for an operator * if

aa-1 =a-1a=e