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Notation

 

This is the primitive membership relationship. means a is a member of b.

 

means a is not a member if b.  

This is the universal quantifier. is true if and only if holds for every set x.  

This is the existential quantifier. is true if and only if there exists at least one x such that is true.  

This is the universal quantifier restricting x to all elements of the set z.  

This is the existential quantifier restricting x to search for elements in the set z.  

is true if and only if there is one and only one set x such that is true.  

This is logical equivalence. For any two statements A and B is true if and only if A and B are either both true or both false.  

This is logical implication. For any two statements A and B is true if and only if A is false or B is true. In other words if A is true then B must be true.

 

We can write a program to list all the statements in the language of ZF. refers to the nth statement output by such a program.

 

This is the negation of A.

 

means every member of a is A is a member of B and B contains at least one member not in A.

 

means every member of a is A is a member of B. A and B may be the same set.

 

If x is a set then the set containing x is written as .  

If x and y are sets then the union of x and y (the set containing those sets and only those sets in x or y) is .

 

In ZF the empty set is the integer 0. The integer n is the union of the empty set and all integers less than n.

 

is the union of all integers.

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