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Posted 14 March 2007

Subsets and inclusion

Every integer is a rational number.  This means that the sets  and  have a special relationship to each other: every element of  is an element of . This is the relationship captured by the following definition.

Definition: inclusion

For all sets A and B,  if and only if every element of A is also an element of B.

means that for all x, if  then .

This definition gives a rule of inference.  The statement “  ” is read " A is included in  B" or " A is a subset of  B".

Examples

¨

¨

¨  The interval .

Terminology

Notation

The notation for inclusion has gotten  in the last fifty years.  The sad story is discussed in detail in the section on symbols.  Here I give the three symbols used in abstractmath.org:

¨   means that every element of A is an element of B.  In particular for any set A, .

¨   (A is a proper subset of B) means that every element of A is an element of B, but there is at least one element of B that is not an element of A.  For example,  because every integer is a real number but there are real numbers that are not integers.  (See proper for some ambiguity in the use of this word.)

¨   is the negation of .  It means that there is at least one element of A that is not an element of B.  For example,

“Contain”

The word "contain" is ambiguous as mathematicians usually use it.

¨  If  you may say " A contains  x".

¨  If  you may say “A contains B”.

This can be confusing if the set A contains both sets and other things as elements.   writers say “A contains  x as an element” and “A contains B as a subset”.

Basic facts about inclusion

Fact   For every set A, .   In other words, every set is a subset of itself.

Fact   If  and  then .

Fact   For every set A, Proof.

Warning

The statement that every set is a subset of itself can cause cognitive dissonance.

Assertions form subsets

If P(x) is an assertion whose only variable is x then the set of elements of a set S for which P(x)  is true is a subset of S.  Using setbuilder notation, this subset is .

Examples

¨  Let S = {2, 3, 4, 5 ,6} and let P(n) be the assertion “n is an even integer”.  Then P determines the subset {2, 4, 6} of S, and we could write the subset as .

¨  The circle of radius 1 with center at the origin is the subset  of the xy plane.

¨  If x is real, then  is a subset of the set  of real numbers .  Of course it is the empty set.

¨  If x is a complex variable, then the set  is a subset of .  Of course, it is not empty.  In fact  .