Review Answers
No. Values of are or, there is no value of that makes
because
because is prime but not odd.
Any set of points where and are not collinear.
Conditional Statements
Learning Objectives
Recognize if-then statements.
Identify the hypothesis and conclusion of an if-then statement.
Write the converse, inverse, and contrapositive of an if-then statement.
Understand a biconditional statement.
Introduction
In geometry we reason from known facts and relationships to create new ones. You saw earlier that inductive reasoning can help, but it does not prove anything. For that we need another kind of reasoning. Now you will begin to learn about deductive reasoning, the kind of reasoning used throughout mathematics and science.
If-Then Statements
In geometry, and in ordinary life, we often make conditional, or if-then, statements.
Statement 1: If the weather is nice, I’ll wash the car. (“Then” is implied even if not stated.)
Statement 2: If you work overtime, then you’ll be paid time-and-a-half.
Statement 3: If divides evenly into , then is an even number.
Statement 4: If a triangle has three congruent sides, it is an equilateral triangle. (“Then” is implied; this is a definition.)
Statement 5: All equiangular triangles are equilateral. (“If” and “then” are both implied.)
An if-then statement has two parts.
The “if” part is called the hypothesis.
The “then” part is called the conclusion.
For example, in statement 2 above, the hypothesis is “you work overtime.” The conclusion is “you’ll be paid time-and-a-half.”
Look at statement 1 above. Even though the word “then” is not actually present, the statement could be rewritten as: If the weather is nice, then I’ll wash the car. This is the meaning of statement 1. The hypothesis is “the weather is nice.” The conclusion is “I’ll wash the car.”
Statement 5 is a little more complicated. “If” and “then” are both implied without being stated. Statement 5 can be rewritten as: If a triangle is equiangular, then it is equilateral.
What is meant by an if-then statement? Suppose your friend makes the statement in statement 2 above, and adds another fact.
If you work overtime, then you’ll be paid time-and-a-half.
You worked overtime this week.
If we accept these statements, what other fact must be true? Combining these two statements, we can state with no doubt:
You’ll be paid time-and-a-half this week.
Let’s analyze statement 1, which was rewritten as: If the weather is nice, then I’ll wash the car. Suppose we accept statement 1 and another fact: I’ll wash the car.
Can we conclude anything further from these two statements? No. Even if the weather is not nice, I might wash the car. We do know that if the weather is nice I’ll wash the car. We don’t know whether or not I might wash the car even if the weather is not nice.
Converse, Inverse, and Contrapositive of an If-Then Statement
Look at statement 1 above again.
If the weather is nice, then I’ll wash the car.
This can be represented in a diagram as:
If then .
“If then ” is also written as
Notice that conditional statements, hypotheses, and conclusions may be true or false. and the statement “If then ” may be true or false.
In deductive reasoning we sometimes study statements related to a given if-then statement. These are formed by using and their opposites, or negations (“not”). Note that “not ” is written in symbols as .
and can be combined to produce new if-then statements.
The converse of is
The inverse of is
The contrapositive of is
Now let’s go back to statement 1: If the weather is nice, then I’ll wash the car.
Converse
If I wash the car, then the weather was nice.
Inverse
If the weather is not nice, then I won’t wash the car.
Contrapositive
If I don’t wash the car, then the weather is not nice.
Notice that if we accept statement 1 as true, then the converse and inverse may, or may not, be true. But the contrapositive is true. Another way to say this is: The contrapositive is logically equivalent to the original if-then statement. In future work you may be asked to prove an if-then statement. If it’s easier to prove the contrapositive, then you can do this since the statement and its contrapositive are equivalent.
Example 1
Statement:
If then . True.
Converse:
If , then . False.
A counterexample is , where but is not
Inverse:
If is not then is not . False.
A counterexample is where is not but =
Contrapositive:
If is not , then is not . True.
If is not , then and is not
Example 2
Statement: If then is the midpoint of . False (as shown below).
Needs
Converse: If is the midpoint of , then . True.
Inverse: If , then is not the midpoint of . True.
Contrapositive: If is not the midpoint of , then False (see the diagram above).
Biconditional Statements
You recall that the converse of “If then ” is “If then .” When these two are combined, we have a biconditional statement.
Biconditional: and
In symbols, this is written as:
We read as: “ if and only if ”
Example 3
True statement: if and only if is an obtuse angle.
You can break this down to say:
If then is an obtuse angle and if is an obtuse angle then .
Notice that both parts of this biconditional are true; the biconditional itself is true.
You most likely recognize this as the definition of an obtuse angle.
Geometric definitions are biconditional statements that are true.
Example 4
Let be
Let be
a. Is true?
Yes.
is if then
From algebra we know that if then and If , then we know that
So if then , or is true.
b. Is true?
No.
is if , then
From algebra we know that if , then
However, does not guarantee that
can be less than but still not less than , for example if is .
So if , then , or , is false.
c. Is true?
No.
is if and only if
We saw above that the if part of this statement, which is
If then
This statement is false. One counterexample is .
Note that if either or is false, then is false.
Lesson Summary
In this lesson you have learned how to express mathematical and other statements in if-then form. You also learned that each if-then statement is linked to variations on the basic theme of “If then .” These variations are the converse, inverse, and contrapositive of the if-then statement. Biconditional statements combine the statement and its converse into a single “if and only if” statement. Definitions are an important type of biconditional, or if-and-only-if, statement.
Points to Consider
We called points, lines, and planes the building blocks of geometry. We will soon see that hypothesis, conclusion, as well as if-then and if-and-only-if statements are the building blocks that deductive reasoning, or logic, is built on. This type of reasoning will be used throughout your study of geometry. In fact, once you understand logical reasoning you will find that you apply it to other studies and to information you encounter all your life.
Write the hypothesis and the conclusion for each statement.
If divides evenly into , then is an even number.
If a triangle has three congruent sides, it is an equilateral triangle.
All equiangular triangles are equilateral.
What is the converse of the statement in exercise 1 above? Is the converse true?
What is the inverse of the statement in exercise 2 above? Is the inverse true?
What is the contrapositive of the statement in exercise 3? Is the contrapositive true?
The converse of a statement about collinear points , , and is: If and , then is the midpoint of . What is the statement?
Is it true?
What is the inverse of the inverse of if then ?
What is the one-word name for the converse of the inverse of an if-then statement?
What is the one-word name for the inverse of the converse of an if-then statement?
For each of the following biconditional statements:
Write in words.
Write in words.
Is true?
Is true?
Is true?
Note that in these questions, and could be reversed and the answers would be correct.
A U.S. citizen can vote if and only if he or she is or more years old.
A whole number is prime if and only if it is an odd number.
Points are collinear if and only if there is a line that contains the points.
if only if and
Review Answers
Hypothesis: divides evenly into ; conclusion: is an even number.
Hypothesis: A triangle has three congruent sides; conclusion: it is an equilateral triangle.
Hypothesis: A triangle is equiangular; conclusion: the triangle is equilateral.
If is an even number, then divides evenly into . True.
If a triangle does not have three congruent sides, then it is not an equilateral triangle. True.
If a triangle is not equilateral, then it is not equiangular. True.
If is the midpoint of , then and . False ( and could both be , , etc.).
If then
Contrapositive
Contrapositive
he or she is or more years old; a U. S. citizen can vote; is true; is true; is true.
a whole number is an odd number; a whole number is prime; is false; is false; is false.
a line contains the points; the points are collinear; is is true; is true; is true.
and ; ; is true; is false; is false.
Deductive Reasoning
Learning Objectives
Recognize and apply some basic rules of logic.
Understand the different parts that inductive reasoning and deductive reasoning play in logical reasoning.
Use truth tables to analyze patterns of reasoning.
Introduction
You began to study deductive reasoning, or logic, in the last section, when you learned about if-then statements. Now we will see that logic, like other fields of knowledge, has its own rules. When we follow those rules, we will expand our base of facts and relationships about points, lines, and planes. We will learn two of the most useful rules of logic in this section.
Direct Reasoning
We all use logic—whether we call it that or not—in our daily lives. And as adults we use logic in our work as well as in making the many decisions a person makes every day.
Which product should you buy?
Who should you vote for?
Will this steel beam support the weight you place on it?
What will be your company’s profit next year?
Let’s see how common sense leads to the two most basic rules of logic.
Example 1
Suppose Bea makes the following statements, which are known to be true.
If Central High School wins today, they will go to the regional tournament.
Central High School does win today.
Common sense tells us that there is an obvious logical conclusion if these two statements are true:
Central High School will go to the regional tournament.
Example 2
Here are two true statements.
is an odd number.
Every odd number is the sum of an even and an odd number.
Based on only these two true statements, there is an obvious further conclusion:
is the sum of an even and an odd number.
(This is true, since ).
Example 3
Suppose the following two statements are true.
If you love me let me know, if you don’t then let me go. (A country music classic. Lyrics by John Rostill.)
You don’t love me.
What is the logical conclusion?
Let me go.
There are two statements in the first line. The second one is:
If you don’t (love me) then let me go.
You don’t love me is stated to be true in the second line.
Based on these true statements,“Let me go” is the logical conclusion.
Now let’s look at the structure of all of these examples, using the and symbols that we used earlier.
Each of the examples has the same form.
conclusion:
A more compact form of this argument, (logical pattern) is:
To state this differently, we could say that the true statement follows automatically from the true statements and .
This reasoning pattern is one of the basic rules of logic. It’s called the law of detachment.
Law of Detachment
Suppose and are statements. Then given
and
You can conclude
Practice saying the law of detachment like this: “If is true, and is true, then is true.”
Example 4
Here are two true statements.
If and are a linear pair, then .
and are a linear pair.
What conclusion do we draw from these two statements?
The next example is a warning not to turn the law of detachment around.
Example 5
Here are two true statements.
If and are a linear pair, then
and
What conclusion can we draw from these two statements?
None! These statements are in the form
Note that since and , we also know that , but this does not mean that they are a linear pair.
The law of detachment does not apply. No further conclusion is justified.
You might be tempted to conclude that and are a linear pair, but if you think about it you will realize that would not be justified. For example, in the rectangle below and (and , but , and are definitely NOT a linear pair.
Now let’s look ahead. We will be doing some more complex deductive reasoning as we move ahead in geometry. In many cases we will build chains of connected if-then statements, leading to a desired conclusion. Start with a simplified example.
Example 6
Suppose the following statements are true.
1. If Pete is late, Mark will be late.
2. If Mark is late, Wen will be late.
3. If Wen is late, Karl will be late.
To these, add one more true statement.
4. Pete is late.
One clear consequence is: Mark will be late. But make sure you can see that Wen and Karl will also be late.
Here’s a symbolic form of the statements.
Our statements form a “chain reaction.” Each “then” becomes the next “if” in a chain of statements. The chain can consist of any number of connected statements. Once we add the true statement as above, we know that the conclusion (the then part) of the last statement is justified.
Another way to look at this is to imagine a chain of dominoes. The dominoes are the linked if-then statements. Once the first domino falls, each domino knocks the next one over, and the last domino falls. is the tipping over of the first domino. The final conclusion of t
he last if-then statement is the last domino.
This is called the law of syllogism. A formal statement of this rule of logic is given below.
Law of Syllogism
Suppose , and are statements. Then given that is true and that you have the following relationship:
Then, you can conclude
Inductive vs. Deductive Reasoning
You have now worked with both inductive and deductive reasoning. They are different but not opposites. In fact, they will work together as we study geometry and other mathematics.
How do these two kinds of reasoning complement (strengthen) each other? Think about the examples you saw earlier in this chapter.
Inductive reasoning means reasoning from examples. You may look at a few examples, or many. Enough examples might make you suspect that a relationship is true always, or might even make you sure of this. But until you go beyond the inductive stage, you can’t be absolutely sure that it is always true.
That’s where deductive reasoning enters and takes over. We have a suggestion arrived at inductively. We then apply rules of logic to prove, beyond any doubt, that the relationship is true always. We will use the law of detachment and the law of syllogism, and other logic rules, to build these proofs.
Symbolic Notation and Truth Tables
Logic has its own rules and symbols. We have already used letters like and to represent statements: for the negation (“not”), and the arrow to indicate if-then. Here are two more symbols we can use.
Truth tables are a way to analyze statements in logic. Let’s look at a few simple truth tables.
Example 1
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