How to Write Definitions as Biconditional Statements

TenMarks teaches you how to write definitions as biconditional statements.

How to Write Definitions as Biconditional Statements In this lesson, let's learn about how to write definitions as biconditional statements. Now in geometry, we use biconditional statements to write definitions because it basically proves that what we are saying is indeed absolutely true. So, we’ll use two examples to make our point. First, let's write the definition as a biconditional for a quadrilateral, a four-sided polygon. So let's first understand what is the conditional statement here. The conditional statement as we know has got a hypothesis and a conclusion. The hypothesis in this case is a figure is a quadrilateral. The conclusion, if it is quadrilateral then it has four sides. So if the figure is a quadrilateral, it has four sides. That’s the hypothesis and the conclusion. Now let's figure out if the converse—what would the converse be? The hypothesis for the converse would be if a figure has four sides, then the conclusion must be that it is a quadrilateral. So this is the conditional statement and this is the converse. So, if for both of these to be true, that’s a true biconditional, the way I would write this would be a quadrilateral—we’ll combine both of these. We will basically say a figure is a quadrilateral if and only if it has four sides. This is our P, this is our Q. P says a figure is a quadrilateral, Q says it has four sides. P implies Q, and Q implies P. So, a figure is a quadrilateral if and only if it has four sides, which means if it does not have four sides then it is not a quadrilateral. This is our biconditional and that’s a definition. Definitions of geometry will usually use if and only if. That’s our definition written as a biconditional. Let's use a slightly more difficult example. Use the diagram to write a definition for this figure. What do we see here? What we’re seeing here is an equilateral triangle with three equal sides. So it basically says if the figure is equilateral triangle, it has three equal sides. The other part, the converse of this is if a figure has three equal sides, it is an equilateral triangle. As we can see, if a figure has three equal sides, it's an equilateral triangle. If a figure is an equilateral triangle, it has three equal sides. So based on these, this is the conditional, this is the converse. Based on these, we can write the biconditional of the definition which is a figure is an equilateral triangle if and only if it has three equal sides. A key thing to remember is the conditional and its converse both have to be true. We usually write the definition with first statement if and only if the second statement.