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 $A$ ”