Consider Theorem 3.25:
Theorem 3.25. Let f : A → B, let S, T ⊆ A,...
Consider Theorem 3.25:
Theorem 3.25. Let f : A → B, let S, T ⊆ A, and let V , W
⊆ B.
1. f(S ∪T) = f(S)∪f(T)
2. f(S ∩T) ⊆ f(S)∩f(T)
3. f-1(V ∪W) = f-1(V
)∪f−1(W)
4. f-1(V ∩W) = f-1(V
)∩f−1(W)
(a) Prove statement (2).
(b) Give an explicit example where the two sides are not
equal.
(c) Prove that if f is one-to-one then the two sides must be
equal.
Let a < b, a, b, ∈ R, and let f : [a, b] → R...
Let a < b, a, b, ∈ R, and let f : [a, b] → R be continuous
such that f is twice differentiable on (a, b), meaning f is
differentiable on (a, b), and f' is also differentiable on (a, b).
Suppose further that there exists c ∈ (a, b) such that f(a) >
f(c) and f(c) < f(b).
prove that there exists x ∈ (a, b) such that f'(x)=0.
then prove there exists z ∈ (a, b) such...
Let A, B, C be sets and let f : A → B and g :...
Let A, B, C be sets and let f : A → B and g : f (A) → C be
one-to-one functions. Prove that their composition g ◦ f , defined
by g ◦ f (x) = g(f (x)), is also one-to-one.
Let f : A → B, g : B → C be such that g ◦...
Let f : A → B, g : B → C be such that g ◦ f is one-to-one (1 :
1).
(a) Prove that f must also be one-to-one (1 : 1).
(b) Consider the statement ‘g must also be one-to-one’. If it is
true, prove it. If it is not, give a counter example.
Let S = {a,b,c,d,e,f,g} and let T = {1,2,3,4,5,6,7,8}.
a. How many different functions are there from...
Let S = {a,b,c,d,e,f,g} and let T = {1,2,3,4,5,6,7,8}.
a. How many different functions are there from S to
T?
b. How many different one-to-one functions are there from S to
T?
c. How many different one-to-one functions are there from T to
S?
d. How many different onto functions are there from T to
S?
Let X be a topological space. Let S and T be topologies on X,
where S...
Let X be a topological space. Let S and T be topologies on X,
where S and T are not equal topologies. Suppose (X,S) is compact
and (X,T) is Hausdorff. Prove that T is not contained in S.
Let X be a topological space. Let S and T be topologies on X,
where S...
Let X be a topological space. Let S and T be topologies on X,
where S and T are distinct topologies. Suppose (X,S) is compact and
(X,T) is Hausdorff. Prove that T is not contained in S.
how to solve?