Chapter 2 T/F
(a) If A∩B=∅, then either A=∅ or B=∅.
F
(c) If x∈A\B ,then x∈A or x∉B.
F
A function f:A→B is surjective if dom f = A.
F
A function from A to B is a nonempty relation f ⊆A×B such that if (a,b)∈ f and(a,c)∈ f, then b=c.
F
A nonempty set S is countable iff there exists an injection f : → S.
F
A relation between A and B is an ordered subset of A × B.
F
A set S is denumerable if there exists a bijection f : → S.
F
Every subset of a denumerable set is denumerable.
F
If R is a relation on S, then {y ∈ S: yRx} determines a partition of S.
F
If a cardinal number is not finite, it is said to be infinite.
F
If a set S is finite, then S is equinumerous with In for some n ∈ .
F
If f:A→B and D is a nonempty subset of B, then f^-1(D) is a non-empty subset of A.
F
If f:A→B, then A is the domain of f and B is the range of f.
F
Iff:A→B is surjective and y∈B, then f^-1(y)∈A.
F
In any relation R on a setS, we always have xRx for all x∈S.
F
In proving S ⊆ T, one should avoid beginning with "Let x ∈ S," because this assumes that S is nonempty.
F
Russell's paradox conflicts with the axiom of separation.
F
The Banach−Tarski paradox uses the axiom of choice.
F
The continuum hypothesis says that א0 is the smallest transfinite cardinal number
F
The identity function maps onto {1}.
F
The set of real numbers is denumerable.
F
The symbol is used to denote the set of all integers.
F
(a,b)=(c,d) iff a=c and b=d.
T
(f) A function f : A → B is bijective if it is one-to-one and maps A onto B.
T
A function f:A→B is injective if for all a and a′ in A, f(a)= f(a′) implies that a = a′.
T
A relation is an equivalence relation if it is reflexive, symmetric, and transitive
T
A slash (/ ) through a symbol means "not."
T
A×B={{a,b}:a∈A and b∈B}.
T
Every subset of a countable set is countable.
T
If f is a function, then the notation y = f (x) means (x, y) ∈ f.
T
If f:A→B and C is a nonempty subset of A, then f(C) is a nonempty subset of B
T
If f:A→B is bijective, then f^-1 :B→A is bijective.
T
If p is a partition of S and x∈S, then x∈A for some A∈p .
T
If p is a partition of set S, then we can obtain a relation R on S by defining x R y iff x and y are in the same piece of the partition.
T
If x∈A∪B, then x∈A or x∈B.
T
IfA⊆BandA≠B,then A is called a proper subset of B.
T
Let S be a nonempty set. There exists an injection f : S → iff there exists a surjection g : → S.
T
The Zermelo−Fraenkel axioms are widely accepted as a foundation for set theory
T
The axiom of regularity rules out the possibility that a set is a member of itself.
T
The composition of two surjective functions is always surjective.
T
The empty set is a subset of every set.
T
The set of rational numbers is denumerable.
T
Two sets S and T are equinumerous if there exists a bijection f : S → T.
T
|S|≤|T| means that there exists an injection f:S→T.
T
