Geometry / Unit 2: Reasoning and Proof
2.2 Biconditionals and Definitions
Notes
A biconditional joins a conditional and its converse with "if and only if" (iff), $p \leftrightarrow q$.
- A biconditional is true only when both the conditional and its converse are true.
- Every good definition can be written as a true biconditional.
- Example: "An angle is right if and only if it measures $90^\circ$." Both directions are true.
Practice
Which statement is a true biconditional?
Only the first is true in both directions. For example, 6 is even but not divisible by 4.
The conditional "If $x = 3$, then $x^2 = 9$" is true. Why is "$x = 3$ if and only if $x^2 = 9$" false?
Both directions must be true for a biconditional.
Which two statements combine to form the biconditional "$p$ if and only if $q$"?
A biconditional is a conditional together with its converse.
Ready for more? Take the Unit 2 test.