Topology
Prove or disprove ( with a counterexample)
(a) The continuous image of a Hausdorff space is Hausdorff.
(b) The continuous image of a connected space is
connected.
Let X, Y be two topological spaces. Prove that if both are T1 or
T2 then X × Y is the same in the product topology. Prove or find a
counterexample for T0.
2. Prove the following properties.(b) Prove that x + ¯ xy = x + y.3. Consider the following Boolean function: F = x¯ y + xy¯ z +
xyz(a) Draw a circuit diagram to obtain the output F. (b) Use the
Boolean algebra theorems to simplify the output function F into the
minimum number of input literals.
Give an example of proof by construction.
For example, prove that for every well-formed formula f in
propositional logic, an equivalent WFF exists in disjunctive normal
form (DNF).
HINT: Every WFF is equivalent to a truth function, and we can
construct an equivalent WFF in full DNF for every truth function.
Explain how.