MandysNotes

Tuesday, 18 March 2014 18:08

Equivalence

By  Gideon
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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.

 AequivB

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.

 

Read 1430 times Last modified on Tuesday, 18 March 2014 18:31
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