Half-adder and full-adder using logisim software

The half-adder 

A B S cin

0 0 0 0

0 1 1 0

1 0 1 0

1 1 0 1 

S is 1 when there is an odd number of 1s.=>A XOR B

cin is 1 only when both inputs are 1.=>A AND B

The full-adder

A B Cin S C

0 0 0 0 0

0 0 1 1 0

0 1 0 1 0

0 1 1 0 1

1 0 0 1 0

1 0 1 0 1

1 1 0 0 1

1 1 1 1 1

Simplifying for S using K map and boolean algebra,

we get (A XOR B) XOR Cin for S

and Cin(A XOR B) OR AB