# MandysNotes

Tuesday, 18 March 2014 18:08

## Equivalence

By  Gideon
Rate this item

Now suppose that we know that  A implies B, and that  B implies A

In symbols this is:

$\left( A \rightarrow B \right) \cap \left( B \rightarrow A \right)$

$= \left( \left( \lnot A \right) \cup B \right) \cap \left( \left( \lnot B \right) \cup A \right)$

$A \leftrightarrow B.$

In this case we say that A is equivalent to B.

This relation is symmetric, so B is also equivalent to A.

In words, we would say that an animal is in the set $$A \leftrightarrow B$$ fs that animal is both (either not duck-billed or a mammal) and (not a mammal or duck-billed).

Note that if it is duck-billed then it must be a mammal;

if it is a mammal then it must be duck-billed.

The relation $$A \leftrightarrow B$$ can also be expressed as:

A if and only if B;

B if and only if A.