Proving a problem is NP - complete by reduction from Vertex-cover.

Please see the attachment below for the complete question. We need to prove that the problem is NP -complete by reduction from Vertex-cover. Problem :- Given a collection of sets { S1, S2 ,..., Sn} and a positive integer K. Does there exist a set T with at most K elements such that T disjoint Si not-equalto empty , 1<=i<=n ? [we can use the edges in graph to create the sets in the collection ]

Understanding a binary search.

What is a binary search, and how does it work?

Prove a tautology using truth tables.

Show that [(a OR b) AND (a IMPLIES c) AND (b IMPLIES c)] IMPLIES c) is a tautology, using truth tables.

Determining the greatest common divisor using the Euclidean Algorithm.

How do I use the Euclidean Algorithm for determining the greatest common divisor of two integers?

Writing regular expressions describing languages of strings.

Write a regular expression describing the language of strings that do not begin with the pattern 110. The language consists of 0s and 1s.