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.
L ={x^a y^b z^c | c=a+b}
a) Prove that L is not regular.
b)Prove by giving a context-free grammar that the L is context
free.
c)Give a regular expression of the complement L'.
L ={x^a y^b z^c | c=a+b} a) Prove that L is not regular. b)Prove
by giving a context-free grammar that the L is context free. c)Give
a regular expression of the complement L'.