Problem Solving
Careful Use of Language in Mathematics
This section might seem like a bit of a sidetrack from the idea of problem solving, but in fact it is not. Mathematics is a social endeavor. We do not just solve problems and then put them aside. Problem solving has (at least) three components:
- Solving the problem. This involves a lot of scratch paper and careful thinking.
- Convincing yourself that your solution is complete and correct. This involves a lot of self-check and asking yourself questions.
- Convincing someone else that your solution is complete and correct. This usually involves writing the problem up carefully or explaining your work in a presentation.
If you are not able to do that last step, then you have not really solved the problem. We will talk more about how to write up a solution soon. Before we do that, we have to think about how mathematicians use language (which is, it turns out, a bit different from how language is used in the rest of life).
Mathematical Statements
Definition
A mathematical statement is a complete sentence that is either true or false, but not both at once.
So a “statement” in mathematics cannot be a question, a command, or a matter of opinion. It is a complete, grammatically correct sentence (with a subject, verb, and usually an object). It is important that the statement is either true or false, though you may not know which! (Part of the work of a mathematician is figuring out which sentences are true and which are false.)
Think / Pair / Share
For each English sentence below, decide if it is a mathematical statement or not. If it is, is the statement true or false (or are you unsure)? If it is not a mathematical statement, in what way does it fail?
- Blue is the prettiest color.
- 60 is an even number.
- Is your dog friendly?
- Honolulu is the capital of Hawaii.
- This sentence is false.
- All roses are red.
- UH Manoa is the best college in the world.
- 1/2 = 2/4.
- Go to bed.
- There are a total of 204 squares on an 8 × 8 chess board.
Now write three mathematical statements and three English sentences that fail to be mathematical statements.
Notice that “1/2 = 2/4” is a perfectly good mathematical statement. It does not look like an English sentence, but read it out loud. The subject is “1/2.” The verb is “equals.” And the object is “2/4.” This is a very good test when you write mathematics: try to read it out loud. Even the equations should read naturally, like English sentences.
Statement (5) is different from the others. It is called a paradox: a statement that is self-contradictory. If it is true, then we conclude that it is false. (Why?) If it is false, then we conclude that it is true. (Why?) Paradoxes are no good as mathematical statements, because it cannot be true and it cannot be false.
And / or
Consider this sentence:
After work, I will go to the beach, or I will do my grocery shopping.
In everyday English, that probably means that if I go to the beach, I will not go shopping. I will do one or the other, but not both activities. This is called an “exclusive or.”
We can usually tell from context whether a speaker means “either one or the other or both,” or whether he means “either one or the other but not both.” (Some people use the awkward phrase “and/or” to describe the first option.)
Remember that in mathematical communication, though, we have to be very precise. We cannot rely on context or assumptions about what is implied or understood.
Definition
In mathematics, the word “or” always means “one or the other or both.”
The word “and” always means “both are true.”
Think / Pair / Share
For each sentence below:
- Decide if the choice x = 3 makes the statement true or false.
- Choose a different value of that makes the statement true (or say why that is not possible).
- Choose a different value of that makes the statement false (or say why that is not possible).
- x is odd or x is even.
- x is odd and x is even.
- x is prime or x is odd.
- x > 5 or x < 5.
- x > 5 and x < 5.
- x + 1 = 7 or x – 1 = 7.
- x·1 = x or x·0 = x.
- x·1 = x and x·0 = x.
- x·1 = x or x·0 = 0.
Quantifiers
Problem 23 (All About the Benjamins)
You are handed an envelope filled with money, and you are told “Every bill in this envelope is a $100 bill.”
- What would convince you beyond any doubt that the sentence is true? How could you convince someone else that the sentence is true?
- What would convince you beyond any doubt that the sentence is false? How could you convince someone else that the sentence is false?
Suppose you were given a different sentence: “There is a $100 bill in this envelope.”
- What would convince you beyond any doubt that the sentence is true? How could you convince someone else that the sentence is true?
- What would convince you beyond any doubt that the sentence is false? How could you convince someone else that the sentence is false?
Think / Pair / Share
What is the difference between the two sentences? How does that difference affect your method to decide if the statement is true or false?
Some mathematical statements have this form:
- “Every time…”
- “For all numbers. . . ”
- “For every choice. . . ”
- “It’s always true that. . . ”
These are universal statements. Such statements claim that something is always true, no matter what.
- To prove a universal statement is false, you must find an example where it fails. This is called a counterexample to the statement.
- To prove a universal statement is true, you must either check every single case, or you must find a logical reason why it would be true. (Sometimes the first option is impossible, because there might be infinitely many cases to check. You would never finish!)
Some mathematical statements have this form:
- “Sometimes…”
- “There is some number. . . ”
- “For some choice. . . ”
- “At least once…”
These are existential statements. Such statements claim there is some example where the statement is true, but it may not always be true.
- To prove an existential statement is true, you may just find the example where it works.
- To prove an existential statement is false, you must either show it fails in every single case, or you must find a logical reason why it cannot be true. (Sometimes the first option is impossible!)
Think / Pair / Share
For each statement below, do the following:
- Decide if it is a universal statement or an existential statement. (This can be tricky because in some statements the quantifier is “hidden” in the meaning of the words.)
- Decide if the statement is true or false, and do your best to justify your decision.
- Every odd number is prime.
- Every prime number is odd.
- For all positive numbers .
- There is some number such that .
- The points (1, 1), (2, 1), and (3, 0) all lie on the same line.
- Addition (of real numbers) is commutative.
- Division (of real numbers) is commutative.
Look back over your work. you will probably find that some of your arguments are sound and convincing while others are less so. In some cases you may “know” the answer but be unable to justify it. That is okay for now! Divide your answers into four categories:
- I am confident that the justification I gave is good.
- I am not confident in the justification I gave.
- I am confident that the justification I gave is not good, or I could not give a justification.
- I could not decide if the statement was true or false.
Conditional Statements
Problem 24 (Card Logic)
You have a deck of cards where each card has a letter on one side and a number on the other side. Your friend claims: “If a card has a vowel on one side, then it has an even number on the other side.”
These cards are on a table.
Which cards must you flip over to be certain that your friend is telling the truth?
Think / Pair / Share
After you have thought about the problem on your own for a while, discuss your ideas with a partner. Do you agree on which cards you must check? Try to come to agreement on an answer you both believe.
Here is another very similar problem, yet people seem to have an easier time solving this one:
Problem 25 (IDs at a Party)
You are in charge of a party where there are young people. Some are drinking alcohol, others soft drinks. Some are old enough to drink alcohol legally, others are under age. You are responsible for ensuring that the drinking laws are not broken, so you have asked each person to put his or her photo ID on the table. At one table, there are four young people:
- One person has a can of beer, another has a bottle of Coke, but their IDs happen to be face down so you cannot see their ages.
- You can, however, see the IDs of the other two people. One is under the drinking age, the other is above it. They both have fizzy clear drinks in glasses, and you are not sure if they are drinking soda water or gin and tonic.
Which IDs and/or drinks do you need to check to make sure that no one is breaking the law?
Think / Pair / Share
After you have thought about the problem on your own for a while, discuss your ideas with a partner. Do you agree on which cards you must check? Compare these two problems. Which question is easier and why?
Definition
A conditional statement can be written in the form
If some statement then some statement.
Where the first statement is the hypothesis and the second statement is the conclusion.
Think / Pair / Share
These are each conditional statements, though they are not all stated in “if/then” form. Identify the hypothesis of each statement. (You may want to rewrite the sentence as an equivalent “if/then” statement.)
- If the tomatoes are red, then they are ready to eat.
The tomatoes are red. / The tomatoes are ready to eat. - An integer n is even if it is a multiple of 2.
n is even. / n is a multiple of 2. - If n is odd, then n is prime.
n is odd. / n is prime. - The team wins when JJ plays.
The team wins. / JJ plays.
Remember that a mathematical statement must have a definite truth value. It is either true or false, with no gray area (even though we may not be sure which is the case). How can you tell if a conditional statement is true or false? Surely, it depends on whether the hypothesis and the conclusion are true or false. But how, exactly, can you decide?
The key is to think of a conditional statement like a promise, and ask yourself: under what condition(s) will I have broken my promise?
Examples
Example 1. Here is a conditional statement:
If I win the lottery, then I’ll give each of my students $1,000.
There are four things that can happen:
- True hypothesis, true conclusion: I do win the lottery, and I do give everyone in class $1,000. I kept my promise, so the conditional statement is TRUE.
- True hypothesis, false conclusion: I do win the lottery, but I decide not to give everyone in class $1,000. I broke my promise, so the conditional statement is FALSE.
- False hypothesis, true conclusion: I do not win the lottery, but I am exceedingly generous, so I go ahead and give everyone in class $1,000. I did not break my promise! (Do you see why?) So the conditional statement is TRUE.
- False hypothesis, false conclusion: I do not win the lottery, so I do not give everyone in class $1,000. I did not break my promise! (Do you see why?) So the conditional statement is TRUE.
What can we conclude from this? A conditional statement is false only when the hypothesis is true and the conclusion is false. In every other instance, the promise (as it were) has not been broken. If a mathematical statement is not false, it must be true.
Example 2. Here is another conditional statement:
If you live in Honolulu, then you live in Hawaii.
Is this statement true or false? It seems like it should depend on who the pronoun “you” refers to, and whether that person lives in Honolulu or not. Let us think it through:
- Sookim lives in Honolulu, so the hypothesis is true. Since Honolulu is in Hawaii, she does live in Hawaii. The statement is true about Sookim, since both the hypothesis and conclusion are true.
- DeeDee lives in Los Angeles. The statement is true about DeeDee since the hypothesis is false.
So in fact it does not matter! The statement is true either way. The right way to understand such a statement is as a universal statement: “Everyone who lives in Honolulu lives in Hawaii.”
This statement is true, and here is how you might justify it: “Pick a random person who lives in Honolulu. That person lives in Hawaii (since Honolulu is in Hawaii), so the statement is true for that person. I do not need to consider people who do not live in Honolulu. The statement is automatically true for those people, because the hypothesis is false!”
Example 3. How do we show a (universal) conditional statement is false?
You need to give a specific instance where the hypothesis is true and the conclusion is false. For example:
If you are a good swimmer, then you are a good surfer.
Do you know someone for whom the hypothesis is true (that person is a good swimmer) but the conclusion is false (the person is not a good surfer)? Then the statement is false!
Think / Pair / Share
For each conditional statement, decide if it is true or false. Justify your answer.
- If then .
- If then .
- If then all odd numbers are prime.
- If then all odd numbers are prime.
- If a number has a 4 in the one’s place, then the number is even.
- If a number is even, then the number has a 4 in the one’s place.
- If the product of two numbers is 0, then one of the numbers is 0.
- If the sum of two numbers is 0, then one of the numbers is 0.
Think / Pair / Share (Two truths and a lie)
On your own, come up with two conditional statements that are true and one that is false. Share your three statements with a partner, but do not say which are true and which is false. See if your partner can figure it out!