18 Sets
Nicholas Bourbaki was a pseudonym for a self-selected group of French academics who
wanted to rewrite mathematics from the bottom up in ‘the right way’. Their bold claim
was that everything should be based on the theory of sets. The axiomatic method was
central and the books they put out were written in the rigorous style of ‘definition,
theorem and proof’. This was also the thrust of the modern mathematics movement of
the 1960s.
Georg Cantor created set theory out of his desire to put the theory of real
numbers on a sound basis. Despite initial prejudice and criticism, set theory was
well established as a branch of mathematics by the turn of the 20th century.
The union of A and BWhat are sets?
A set may be regarded as a collection of objects. This is informal but gives us
the main idea. The objects themselves are called ‘elements’ or ‘members’ of the
set. If we write a set A which has a member a, we may write a ∈ A, as did
Cantor. An example is A = {1, 2, 3, 4, 5} and we can write 1 ∈ A for
membership, and 6 ∈ A for non-membership.
Sets can be combined in two important ways. If A and B are two sets then the
set consisting of elements which are members of A or B (or both) is called the
‘union’ of the two sets. Mathematicians write this as A ∪ B. It can also be
described by a Venn diagram, named after the Victorian logician the Rev. John