A biconditional statement combines a conditional statement with its converse statement. Both the conditional and converse statements must be true to produce a biconditional statement. If we remove the if-then part of a true conditional statement, combine the hypothesis and conclusion, and tuck in a phrase "if and only if," we can create biconditional statements. Geometry and logic cross paths many ways. One example is a biconditional statement. To understand biconditional statements, we first need to review conditional and converse statements. Then we will see how these logic tools apply to geometry.
Get free estimates from math tutors near you.Each of these conditional statements has a hypothesis ("If …") and a conclusion (" …, then …").
These statements can be true or false. Whether the conditional statement is true or false does not matter (well, it will eventually), so long as the second part (the conclusion) relates to, and is dependent on, the first part (the hypothesis).
To create a converse statement for a given conditional statement, switch the hypothesis and the conclusion. You may "clean up" the two parts for grammar without affecting the logic.
Take the first conditional statement from above:
Create the converse statement:
This converse statement is not true, as you can conceive of something … or someone … else eating your homework: your dog, your little brother. Your homework being eaten does not automatically mean you have a goat.
Let's apply the same concept of switching conclusion and hypothesis to one of the conditional geometry statements:
This converse is true; remember, though, neither the original conditional statement nor its converse have to be true to be valid, logical statements.
For, "If the polygon has only four sides, then the polygon is a quadrilateral," write the converse statement.
Converse: If the polygon is a quadrilateral, then the polygon has only four sides.
Try this one, too: "If the quadrilateral has four congruent sides and angles, then the quadrilateral is a square."
Converse: If the quadrilateral is a square, then the quadrilateral has four congruent sides and angles.
The general form (for goats, geometry or lunch) is:
Hypothesis if and only if conclusion.
Because the statement is biconditional (conditional in both directions), we can also write it this way, which is the converse statement:
Conclusion if and only if hypothesis.
Notice we can create two biconditional statements. If conditional statements are one-way streets, biconditional statements are the two-way streets of logic.
Both the conditional and converse statements must be true to produce a biconditional statement.
Since both statements are true, we can write two biconditional statements:
You can do this if and only if both conditional and converse statements have the same truth value. They could both be false and you could still write a true biconditional statement ("My pet goat draws polygons if and only if my pet goat buys art supplies online.").
Let's see how different truth values prevent logical biconditional statements, using our pet goat:
We can attempt, but fail to write, logical biconditional statements, but they will not make sense: