- 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
**n**th 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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