In this unit, we will learn the difference between inductive and deductive reasoning. You will make conjectures and learn how to verify them with the use of deductive reasoning. In addition to the various conditional statements, we will also explore biconditional statements.
A conclusion based on a pattern is called a conjecture. A conjecture is a guess based on analyzing information or observing a pattern. When we come to a conclusion or make a general rule based on a pattern that we observe, we are using inductive reasoning.
Since a conjecture is an educated guess, it may be true or false. A conjecture is said to be false if there is even one situation in which the conjecture is not true. We call the false example a counterexample.
When we make a conclusion after examining several specific cases, we have used inductive reasoning. However, we must be cautious -- by finding only one counterexample, we can disprove the conclusion.
Related Conditional Statements
If you change the hypothesis or conclusion of a conditional statement, you form a related conditional. There are three related conditionals -- converse, inverse, and contrapositive.
A conditional consists of a hypothesis (p) and a conclusion (q).
A conditional statement is a statement that can be written as an if-then statement -- "if p, then q" or "p implies q." An if-then statement is a statement such as "If two angles are vertical angles, then they are congruent." The phrase immediately following the word "if" is the hypothesis -- two angles are vertical angles. The phrase immediately following the word "then" is the conclusion -- they are congruent.
Sometimes it is necessary to rewrite a conditional statement so that it is in the "if-then" form. Here is an example:
Conditional: A person who does his homework will improve his math grade.
If-Then Form: If a person does his homework, then he will improve his math grade.
A conditional statement has a false truth value ONLY if the hypothesis (p) is true and the conclusion (q) is false. A conditional statement always has the same truth value as its contrapositive, and the converse and inverse always have the same truth value.
The converse statement is formed by exchanging the hypothesis with the conclusion.
Let's use our original conditional statement -- "If two angles are vertical angles, then they are congruent." To write the converse, we simply exchange the hypothesis (two angles are vertical angles) with the conclusion (they are congruent). Our converse statement becomes
If two angles are congruent, then they are vertical angles.
The inverse statement is formed by negating the hypothesis and the conclusion. The negation of a statement, "not p," has the opposite truth value of the original statement.
If p is true, then not p is false.
If p is false, then not p is true.
Once again, we will use our original conditional statement, "If two angles are vertical angles, then they are congruent." The inverse (negate both the hypothesis and the conclusion) is as follows
If two angles are not vertical angles, then they are not congruent.
The contrapositive is formed by doing both -- exchanging and negating the hypothesis and the conclusion.
If two angles are not congruent, then they are not vertical angles.
With inductive reasoning, we use examples to make conjecture. With deductive reasoning, we use facts, definitions, rules, and properties to draw conclusions and prove that our conjectures are true.
Law of Detachment
One form of deductive reasoning that lets us draw conclusions from true facts is called the Law of Detachment. If the conditional statement (if p then q) is a true statement and the hypothesis (p) is true, then the conclusion/conjecture (q) is true.
Given: If i get over 90%, I will receive an A. I got 96%.
Conjecture: I have an A.
Law of Syllogism
Another form of deductive reasoning is the Law of Syllogism. If two conditional statements (if p then q) and (if q then r) are true, then the resulting conditional statement (if p then r) is also true.
Given: If I oversleep, I will miss the bus. If I miss the bus, I will have to walk to school.
Conjecture: If I oversleep, I will have to walk to school.
A biconditional statement combines a conditional statement, "if p, then q," with its converse, "if q, then p."
Definitions can be written as biconditionals:
Definition: Circumference is the distance around a circle.
Biconditional: A measure is the circumference if and only if it is the distance around a circle.
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