Set Theory: Sets and Functions
Definitions of
Further definitions on
Definitions of (real) function
Theorem 1
Theorem 2
Theorem 3
Note
Definitions of decomposition or partition into classes, a is related to b by the (binary) relation R, equivalence relation, reflexitivity, symmetry, transitivity.
Theorem 4
Problems
Show
and since and since . Hence,Show
Take
to be the empty set and to be any non-empty set.Let
and . Find and .Prove that
andStep through using definitions, complements, idempotence, duality principle, distributivity, associativity, commutativity.
Prove
Evident once some thought has been applied.
Let
be the set of all positive integers divisible by . What are and ?
What are
and ?If
If
Let
be the set of points lying on the curve with . What is ? (fractional part of ). Prove every closed interval of length one has the same image under . What is this image? Is one-to-one? What is the preimage of the interval ? Partition the real line into classes of points with the same image.Simple enough.
Given set
, let be the set of all ordered pairs with . Let . Interpret .Equality.
Find a binary relation which is:
Reflexive and symmetric, but not transitive.
Reflexive, but neither symmetric nor transitive.
Symmetric, but neither reflexive nor transitive.
Transitive, but neither reflexive nor symmetric.
Blood relative. Remembers name of (provided everybody knows their own name). Not equal to. Strictly greater than. (Credit to CoveredInChocolate for the first two answers.)