Conditional and biconditional statements geometry :
In this section, we are going to study a type of logical statement called conditional statement. A conditional statement has two parts, a hypothesis and a conclusion. If the statement is written in if-then form, the "if" part contains the hypothesis and the "then" part contains the conclusion.
Here is an example :
Conditional statements can be either true or false.
To show that a conditional statement is true, we must present an argument that the conclusion follows for all cases that fulfill the hypothesis.
To show that a conditional statement is false, describe a single counter example that shows the statement is not always true.
The converse of a conditional statement is formed by switching the hypothesis and conclusion.
Here is an example.
Statement : If you hear thunder, then you see lightning.
Converse : If you see lightning, then you hear thunder.
A statement can be altered by negation, that is, by writing the negative of the statement.
Here are some examples.
∠A = 65°
∠A is obtuse
∠A ≠ 65°
∠A is not obtuse
When we negate the hypothesis and conclusion of a conditional statement, we form the inverse. When we negate the hypothesis and conclusion of the converse of a conditional statement, we form the contrapositive.
If∠A = 65°, then ∠A is obtuse.
If∠A ≠ 65°, then ∠A is not obtuse.
If∠A is obtuse, ∠A = 65°.
If ∠A is not obtuse, then ∠A ≠ 65°.
In the above,
(i) Both Original and Contrapositive are true.
(ii) Both inverse and converse are false.
When two statements are both true or both false, they are called equivalent statements. A conditional statement is equivalent to its contrapositive. Similarly, the inverse and converse of any conditional statement are equivalent. This is shown above.
All definitions can be interpreted "forward" and "backward". For instance, the definition of perpendicular lines means
(i) If two lines are perpendicular, then they intersect to form a right angle.
(ii) If two lines intersect to form a right angle, then they are perpendicular.
Conditional statements are not always written in if-then form. Another common form of a conditional statement is only-if-form.
Here is an example.
We can rewrite this conditional statement in if-then form as follows :
If it is Sunday, then I am in park.
A biconditional statement is a statement that contains the phrase "if and only if". Writing biconditional statement is equivalent to writing a conditional statement and its converse.
A biconditional statement can be either true or false. To be true,both the conditional statement and its converse must be true. This means that a true biconditional statement is true both “forward” and “backward.” Alldefinitions can be written as true biconditional statements.
To get example problems on Conditional statements,
To get example problems on definitions and biconditional statements,
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