I know that
𝑞 → 𝑝 is the converse of 𝑝 → 𝑞.
¬𝑞→ ¬ 𝑝 is the contrapositive of 𝑝 → 𝑞
¬ 𝑝→ ¬ 𝑞 is the inverse of 𝑝 → 𝑞
However, I'm not sure if it's the same for bidirectional statements. Am I right in the following statements?
Original statement: We’ll win the ICG cup only if we have enough players.
Converse: Only if we have enough players, will we then win the ICG cup.
Contrapositive: Only if we do not have players, will we then not win the ICG cup.
Inverse: We’ll not win the ICG cup, only if we do not have enough players.
If not, how do I go about obtaining the correct statements?
Best Answer
Close. The converse and contrapositive are off. The original statement is not a biconditional statement (those are of the form "X if and only if Y"). You could rewrite any conditional of the form "X only if Y" equivalently as "if X, then Y", that is, $X \rightarrow Y$.
Original statement ($X \rightarrow Y$): "We’ll win the ICG cup only if we have enough players". This is the same as saying, "If we'll win the ICG cup, then we have enough players." In this example X="we'll win the ICG cup" and Y="we have enough players". Plug in the values for your converse/inverse/contrapositive definitions to get the following.
Converse ($Y \rightarrow X$): "We have enough players only if we win the ICG cup." It is equivalent to say, "If we have enough players, then we'll win the ICG cup."
Contrapositive ($\neg Y \rightarrow \neg X$): "We don't have enough players only if we don't win the ICG cup." This is equivalent to saying "If we don't have enough players, then we won't win the ICG cup."
Inverse ($\neg X \rightarrow \neg Y$): "We won't win the ICG cup only if we don't have enough players". Equivalently you could say "If we won't win the ICG cup, then we don't have enough players."