Introduction to Mathematical Thinking Week2
Implication The implication is important in mathematics. $\phi$ implies $\psi$ is denoted as $\phi\implies\psi$, that means the truth of $\psi$ follows from the truth of $\phi$. And $\phi$ is called antecedent and $\psi$ is consequent. The truth table of $\phi\implies\psi$ as follows: $\phi$ $\psi$ $\phi\implies\psi$ T T T T F F F T T F F T The T denotes the truth and F denotes the false. A true conclusion from a true assumption, so the first row is true. If that implication is true, that means $\psi$ would have to be T if $\phi$ is T. So we cannot have $\phi$ is T and $\psi$ is F if $\phi\implies\psi$ is T. Hence $\phi\implies\psi$ must be F. We can look at “$\phi$ does not imply $\psi$” ($\phi\nRightarrow\psi$) that is even through $\phi$ is T, $\psi$ is nevertheless F. So $\phi\nRightarrow\psi$ is T if and only if $\phi$ is T and $\psi$ is F. In all other circumstances, $\phi\nRightarrow\psi$ is F, which means $\phi\implies\psi$ is T. So, the third and fourth rows are T. 📝 Note: The implication involves causality. For example, “$\sqrt{2}$ is irrational” does not imply “$1+1=2$”, because this two statements has no relationship, they are independent of each other. ...