For the following conditional statement write the i contrapo

For the following conditional statement write the i) contrapositive, ii) converse and iii) inverse.

(Don’t forget to use DeMorgan’s laws where needed.)

“If K is a chain complex such that Z_n(K) is a direct summand of K_n for all n and H_n(K) = 0 for all n, then the identity map and the zero map of K into itself are chain homotopic”

Solution

i) Contrapositive

If K is a chain complex such that identity map or the zero map of K into itself are not chain homotopic then there exist an n such that Z_n(K) is not a direct summand of K_n or there exist an n such that H_n(K) /= 0.

ii) Converse

If K is a chain complex such that the identity map and the zero map of K into itself are chain homotopic then Z_n(K) is a direct summand of K_n for all n and H_n(K) = 0 for all n.

iii) Inverse

If K is a chain complex for which there exist an n such that Z_n(K) is not a direct summand of K_n or there exist an n such that H_n(K) /= 0, then the identity map or the zero map of K into itself are not chain homotopic.