Whenever a handshake happens,one of the following 3 cases may happen.
1--2 persons get added into Odd set
2--one person of Odd set get substituted by a new person.
3--2 persons get removed from the group.
EXAMPLE:
Consider there are 4 persons a,b,c,d in party.
Let us see what happens for every hand shake.
1--a,b shake hands. Now Odd={a,b} ,even={}
2--c,d shake hands.Now Odd={a,b,c,d},even={}.
3--a,b shake hands again.Now Odd={c,d},even={a,b}.Since a,b had 2 handshakes each and c,d had 1 hand shake each.
4--a,c shake hands.Now Odd={a,d},even={c,b}.since a had 3 hand shakes,d had 1 hand shake ,c and b had 2 handshakes each.
Therefore we can say that after every hand shake, cardinality of Odd set will either increase by 2 or remains same or decrease by 2. So cardinality of Odd set is always even .
Hence option A is true.
you can see other options value may change for a new handshake.
Ps:
If 2 persons in Odd set shake hand ,then both of them move into Even set.
If 2 persons in Even set shake hand,then both of them move into Odd set.
If a person x in Odd set and person y in even set shake hand,then x moves into Odd set and y moves into Even set.