Mathematical implication

Summary

• If \(A\) and \(B\) are two predicates , then the statement “ If \(A\) , then \(B\) ” is called an implication .
• Example: “If \(x=2\) then \(x^2=4"\) . This implication is a true statement.
• \(A\) is called the hypothesis and \(B\) the conclusion of the implication .
• An implication is true unless you can find a value for the free variable such that the conclusion is false when the hypothesis is true.
• Example: “If \(x^2=4\) then \(x=2\) ” is false. The hypothesis is true if \(x=-2\) .
• Example: “If penguins fly then the moon is made of cheese” is vacuously true since the hypothesis is false.
• Example: “If \(x=x+1\) then \(x=π\) ” is vacuously true as the hypothesis if false for all real numbers.
• Alternative expressions for “If \(A\) , then \(B\) ”:
• “ \(A \implies B\) ”
• “ \(A\) is sufficient for \(B\) ” . Example “ \(x=1 \implies x>0\) ” ( \(A\) is not necessary)
• “ \(B\) is necessary for \(A\) ” . Example “ \(x>1 \implies x>0\) ”
• “ \(B\) is a consequence of \(A\) ”
• “ \(B\) if \(A\) ”
• “ \(B\) is an implication of \(A\) ”