Set Theoery - Terms and Symbols

Mathematics 143
Set Theory - Terms and Symbols

I. Set notations

(a)	Capital letters : A, B, C, S, T
(b)	Curly brackets containing a list of elements. Example : {a, b, c} is the set containing the letters a, b, and c.
(c)	Set-builder notation (see Rolf Pg. 425). Example : {x \| x satisfies property P} where P is any property which makes sense when applied to x. It may be either easy or difficult to determine whether or not property P is true for certain values of x. If the determination takes a very long time, such as a billion years, the set may have uncertain status.

II. Special sets

(a)	U : The UNIVERSE SET, i.e., the set of all things (or elements) under discussion at the moment. U changes from one problem to another.
(b)	Ø or { } : the EMPTY SET or NULL SET, containing no elements. Note that { } is different from the number "0" and the sets { 0 } and { Ø }. A common way to write the number "zero" is to slash it, so that it differs from the letter "O", but this practice must be avoided when the empty set enters a discussion.

III. Relations

(a)

The following 6 statements all mean the same thing : "A

B" , "B

A" ,
"A is contained in B", "B contains A", "A is a subset of B",
and "every element of A is an element of B".

(b)

(c)

or =

B" and "A = B" mean that the sets A and B have precisely the same elements (as well as the same size).

(d)

means "not" when drawn across any symbol representing a verb. For example, A

B means "A is not the same set as B"

(e)

e or

indicates set membership. Thus, "x e A" means that "x is a member of the set A" or "x is an element of the set A".

IV. Operations

(a)	A B = {x \| x e A AND x e B} (See Rolf Pg 428 Fig 6-4)
(b)	A B = {x \| EITHER x e A OR x e B} (See Rolf Pg 427 Fig 6-3)
(c) A' or A^c	A' = {x \| x is not in the set A} (See Rolf Pg 429 Fig 6-6)
(d) n(A) or \|A\|	n(A) = is the size of A, or the number of elements in A.

V. Venn Diagrams

Venn Diagrams are stylized pictures of sets. See exercises