A pair

G=(V,E)

is a directed graph if

V

is a set and

EsubV\\\\times V

. The graph\

G

has a cycle if there exists an integer

n>=1

and elements

v_(1),dots,v_(n)inV

such\ that

(v_(1),v_(2)),(v_(2),v_(3)),dots,(v_(n-1),v_(n)),(v_(n),v_(1))inE

. Show that if

V

is a non-empty\ finite set and for all

vinV

there is

v^(')inV

such that

(v,v^('))inE

, then

G

has a\ cycle.

4. A pair G=(V,E) is a directed graph if V is a set and EVV. The graph G has a cycle if there exists an integer n1 and elements v1,,vnV such that (v1,v2),(v2,v3),,(vn1,vn),(vn,v1)E. Show that if V is a non-empty finite set and for all vV there is vV such that (v,v)E, then G has a cycle