Cardinality of real numbers and set of real numbers between 0 and 1

Question : Show that the set of all real numbers has the same cardinality as the set of real number between 0 and 1.

Solution : Solution Let S be the open interval of real numbers between 0 and 1

S={x  ∈ R| 0 < x < 1 }.

Imagine picking up S and bending it into circle as shown below. Since S does not include either endpoint 0 or 1, the top-most point of the circle is omitted from the drawing.

Continue with reading

Comparison of Dijkstra’s algorithm and Floyd–Warshall algorithm

Floyd-Warshall Algorithm

Stephen Warshall and Robert Floyd independently discovered Floyd’s algorithm in 1962.

The algorithm solves a type of problem call the all-pairs shortest-path problem. Actually, the Warshall version of the algorithm finds the transitive closure of a graph but it does not use weights when finding a path.  The Floyd algorithm is essentially the same as the Warshall algorithm except it adds weight to the distance calculation.

Continue with reading