2-2 BICONDITIONALS AND DEFINITIONS (p. 75-81)
A biconditional statement is a combination of a true conditional statement and its true
converse in one sentence. The term biconditional literally means two conditionals.
To write the biconditional statement more concisely, join the hypothesis and the
conclusion with the phrase “if and only if.” This phrase is inserted in the middle of the
sentence.
Example: Consider the following true conditional statement. Determine its converse. If
the converse is also true, combine the two statements as a true biconditional using the
phrase if and only if.
If x - 3  9, then x  12.
Do 1 on p. 75.
You can also take a biconditional statement and rewrite it as two conditional statements
that are converses of each other. You are basically doing the reverse of what we just did
in the above example.
Example: Write the two statements that form the following biconditional statement.
A number is divisible by 4 if and only if its last two digits are divisible by 4.
Do 2 on p. 76.
A biconditional statement can be symbolized as p  q. Note the double-headed arrow,
which implies that the sentence is true when read left to right and when read right to left.
Do the Investigation: Writing a Definition on p. 76. Students can work within their
cooperative groups to come up with a definition for a polyglob.
Definitions should use clearly understood words and be precise. Moreover, good
definitions are reversible- they can be written as true biconditional statements.
Example: Show that this definition of quadrilateral is reversible. Then, write it as a true
biconditional.
A quadrilateral is a polygon with exactly four sides.
Do 3 on p. 77.
To show that a statement is not a good definition, just find one counterexample.
Example: Is the following statement a good definition? Explain.
A porcupine is an animal that has spiny projections.
Do 4 on p. 77.
Homework p. 78-81: 1,10,14,16,21,23,27,34,38,46,47,50,52,67,69
47. Carla is the oldest and plays the keyboards.
Since Amy does not play the guitar or keyboards, she plays the drums.
Thus, Bob plays the guitar.