Conditional Statements
Conditional Statements - What are they - Video
Definition of Conditional Statements
Definition of Conditional Statement:
Conditional statement is a logical statement that has two parts: a hypothesis and a conclusion, and it can be written in two ways. If we take a look at the example all dogs have 4 legs. We can the subject all dogs and make that our hypothesis, and make the object 4 legs the conclusion in the if-then form.
So we get If an animal is a dog, then it has 4 legs.
Statements Table
Statements Table:
Here we have added converse, inverse, and contrapositive to the mix.
The converse is flipping the hypothesis and the conclusion. So, we get if an animal has 4 legs, then it is a dog. This is false because one counterexample is a cat.
The inverse is negating the original hypothesis and the original conclusion. So, we get if an animal is not a dog, then it does not have 4 legs. This is false because one counterexample is a lion.
The contrapositive is flipping and negating the original hypothesis and conclusion. So, we get if an animal does not have 4 legs, then it is not a dog. This is true.
The converse and the inverse are usually false but they are true that leads to specific case, which we will talk about it in example 2. And the conditional statement and contrapositive are true.
Example 1
Example 1:
We have the statement 41° is acute. So we take the 41° as our hypothesis and acute as the conclusion.
The conditional statement in if-then form is, if m∠A = 41°, then ∠A is acute. This is true.
The converse is flipping the hypothesis and the conclusion. So we get, if ∠A is acute, then m∠A = 41°. This is false because one counterexample is 88° is an acute angle.
The inverse is negating the original hypothesis and the original conclusion. So we get, if m∠A ≠ 41°, then ∠A is not acute. This is false because one counterexample is 42° is not 41° and 42° is an acute angle.
The contrapositive is flipping and negating the original hypothesis and conclusion. So we get, if ∠A is not acute, then m∠A ≠ 41°. This is true.
Example 2
Example 2:
We have the statement 90° is a right angle. So we take the 90° as our hypothesis and a right angle as the conclusion.
The conditional statement in if-then form is, if m∠A = 90°, then ∠A is a right angle.
The converse is flipping the hypothesis and the conclusion. So we get, if ∠A is a right angle, then m∠A = 90°. This is true.
The inverse is negating the original hypothesis and the original conclusion. So we get, if m∠A ≠ 90°, then ∠A is a not a right angle. This is true.
The contrapositive is flipping and negating the original hypothesis and conclusion. So we get, if ∠A is not a right angle, then m∠A ≠ 90°. This is true.
Biconditional Statement
Biconditional Statement:
Example 2 leads us to a biconditional statment since all 4 are true. The definition of a biconditional statement is the conditional statement and its converse are both true. If that is the case, then we can use the phrase if and only if (iff).
We have the statement 90° is a right angle. We can write that statement in a biconditional way using if and only if.
An angle has a measure of 90° if and only if an angle is a right angle.
Since the original statement is a definition we can write it in biconditional form.