# Understanding Conditionals and Biconditionals in Logic

Conditional statements, also known as if-then statements, play a crucial role in logic. They consist of a hypothesis (following "if") and a conclusion (following "then"). By identifying the hypothesis and conclusion, writing conditional statements, evaluating truth values, and exploring converses, one can grasp the fundamental concepts of conditionals and biconditionals.

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**Unit 2**Conditionals, Biconditionals, and Definitions**Conditional Statements**Conditional statements can also be referred to as if-then statements. Example: If you are not completely satisfied, then your money will be refunded. Every conditional has two parts. They are: Hypothesis The part following if Conclusion The part following then**Identifying the Hypothesis and Conclusion**Example 1: If today is the first day of fall, then the month is September. Hypothesis: ______________________________ Conclusion: ______________________________ Example 2: If y 3 = 5, then y = 8. Hypothesis: ______________________________ Conclusion: ______________________________**Writing a Conditional Statement**Write each sentence as a conditional. Example 1: A rectangle has four right angles. _____________________________________________________ Example 2: A tiger is an animal. _____________________________________________________ Example 3: An integer that ends with 0 is divisible by 5. _____________________________________________________**Truth Value**Examples: Conditionals can have a truth value, which is either true or false. To show that a conditional is true, show that every time the hypothesis is true, the conclusion is also true. Show that his conditional is false by finding a counterexample: If it is February, then there are only 28 days in the month. If the name of a state contains the word New, then the state borders an ocean. To show that a conditional is false, you need to find only one counterexample for which the hypothesis is true and the conclusion is false.**Converse**The converse of a conditional switches the hypothesis and the conclusion. Conditional: If p, then q. Symbolic p q Example: Write the converse of the following conditional statement. Conditional: If two lines intersect to form right angles, then they are perpendicular. Converse: If two lines are perpendicular, then they intersect to form right angles. Converse: If q, then p. q p**Converse**OYO Example: Write the converse of the following conditional statement. Conditional: If two lines are not parallel and do not intersect, then they are skew. Converse: ___________________________________ Conditional: If an angle is a straight line, then its measure is 180o. Converse: ___________________________________**Inverse**The inverse of a conditional is when you make the conditional statement negative. (You add not to the hypothesis and conclusion) Conditional: If p, then q. Symbolic p q Example: Write the inverse of the following conditional statement. Conditional: If two lines intersect to form right angles, then they are perpendicular. Inverse: If two lines do not intersect to form right angles, then they are not perpendicular. Inverse: If ~q, then ~p. ~q ~p**Contrapositive**The contrapositive is when you make the converse statement negative. (You add not to the hypothesis and conclusion) Converse: If q, then p. Contrapositive: If ~q, then ~p. Symbolic q p Example: Write the contrapositive of the following conditional statement. Conditional: If two lines intersect to form right angles, then they are perpendicular. Converse: If two lines are perpendicular, then they intersect to form right angles. Contrapositive: If two lines are not perpendicular, then they do not intersect to form right angles. ~q ~p**Example**All obtuse angles have measures greater than 90 degrees. Conditional: ________________________________________________ Converse: ________________________________________________ Inverse: ________________________________________________ Contrapositive: ________________________________________________**Homework:**Practice Page 2-1A #1-8 Page 1-1B #1-9**Biconditionals and**Definitions**Biconditionals**When a conditional and its converse are true, you can combine them as a true biconditional. This is the statement you get by connecting the conditional and its converse, writing it more concisely by joining the two parts with if and only if.**Examples**Conditional: If two angles have the same measure, then the angles are congruent. Converse: If two angles are congruent, then the angles have the same measure. Are these both true? Biconditional: Two angles have the same measure if and only if the angles are congruent.**Examples**Consider this true conditional statement. Write its converse. If the converse is also true, combine the statements as a biconditional. Conditional: If three points are collinear, then they lie on the same plane. Converse: ____________________________________ Are both true? If so write the biconditional statement. ___________________________________________________**Examples**Write two statements that form each biconditional. 1. A line bisects a segment if and only if the line intersects the segment only at its midpoint. 2. Two lines are parallel if and only if they are coplanar and do not intersect.**Definitions**A good definition is a statement that can help you identify or classify an object. A good definition has several important components. A good definition uses clearly understood terms. A good definition is precise. Avoid words such as large, sort of, and some. A good definition is reversible. That means you can write a good definition as a true biconditional.**Write a Definition as a Biconditional**Definition: Perpendicular lines are two lines that intersect to form right angles. Conditional: If two lines are perpendicular, then they intersect to form right angles. Converse: If two lines intersect to form right angles, then they are perpendicular. Biconditional: Two lines are perpendicular if and only if they intersect to form right angles.**Write a Definition as a Biconditional**Definition: A right angle is an angle whose measure is 90o. Conditional: Converse: Biconditional: