Study Guide

Logic and Proof - Building Mathematical Statements

Building Mathematical Statements

Mathematical statements are exactly the same as fashion statements. Except instead of clothes, we have mathematical formulas. Hopefully we won't get chilly walking down the runway.

The simplest kind of mathematical statement is an explanation of how numbers are related. For example, you might say, "x = 5" or, "4 + 7 = 35" or, "58 is the sum of two prime numbers." As you can see, some statements are true, some are false, and some are as clear as a mud smoothie.

What all of these statements have in common is that they can't be split into simpler statements—they are indivisible (with liberty and justice for all). The Greek word for indivisible is atomos, so we call these statements atoms. Unlike the atoms in chemistry, mathematical atoms can make only statements, not bombs.

  • Negations

    One way to modify a statement is to negate it. This just means to say the statement isn't true (which, in case you were wondering, isn't the same as having an argument). For example, the negation of, "Bacon is delicious," would be, "It's not true that bacon is delicious," or more simply, "Bacon is not delicious." That's a silly example to use though, because everyone knows bacon is delicious.

    Mathematical negation differs a bit from our negation in everyday English. For example, if you say to your friend in a heated argument, "I don't think Edward can even compare to that dreamy Jacob," you really mean, "I think Edward is ugly and Jacob is a were-hottie."

    However, when a mathematician says, "I don't think Edward can even compare to that dreamy Jacob," she's actually saying that the two cannot be compared. Who knows what she actually thinks? She may think they're both ugly, or that neither is actually hot, but that each is attractive in their own unique way. (Actually, she's probably busy working on some groundbreaking theorem while everybody around her is pestering her with vampire gossip.)

    The bottom line is that negations don't allow us to conclude very much.

    Even the simple act of negation can get really complicated really quickly. Consider the statement, "This statement is not true." Is it true, or isn't it? If it's true, then we can trust what it says. But…that means that it's false. Similarly, when it's false, that means it's true. There's no way to make any sense of this madness! What ever will we do?

    While this seems like a fluke, it has pestered both linguists and logicians for centuries. Modern mathematicians have developed a formal way of avoiding such conundrums by putting up restrictions on the kind of statements they can make (another victory for censorship). We should probably do the same, since statements like that just cause headaches.

  • Conjunctions

    There are also ways to connect two statements to make a bigger statement, like Legos. One is by using the word "and," and the result is called the conjunction of the two statements. Sort of like the kind in School House Rock, but not exactly. The statement "A and B" is true exactly when both A is true and B is true. From a more pessimistic point of view, "A and B" is false as soon as one of A or B is false.

    Sample Problem

    Sam is 6 feet tall and has brown hair. He also has blue eyes, his favorite character from The Lord of the Rings is Samwise Gamgee, his favorite album is Abbey Road, he prefers red apples to green, he collects pewter figurines with little crystals embedded as dramatic accents, and he wakes up in the morning screaming "Wednesday!" every Wednesday. Is the statement, "Sam is shorter than 6'3" and has brown hair" true?

    Since the statement is a conjunction, we check each atom individually. Does Sam have brown hair? Yes, it says so right in the first line up there. Is Sam shorter than 6'3"? Yes, his height is an even 6 feet. Since both parts are true, we conclude that the whole statement is true.

    Sample Problem

    Is the statement "Sam's favorite character from The Lord of the Rings is not Gandalf the Grey and his favorite album is not Abbey Road" true?

    Again, let's take a look at each atom. The first atom is a negation, saying that it's not true that Sam's favorite character is Gandalf. That's correct, because his favorite LOTR character is Samwise. The next atom is also a negation that says that it's not true that his favorite album is Abbey Road. But Sam's favorite album is Abbey Road. So the negation is wrong. Since the statement is a conjunction, the whole thing is false.

  • Disjunctions

    The other main way of pasting together statements is by using the word "or," forming a disjunction. The statement "A or B" is true when either A is true, B is true, or both are true. This is another way mathematical language differs from common speech.

    When we snidely say, "I can take out the trash or I can wash the dishes," we usually mean we'll do one or the other, but not both. Of course, if you have a mathematician parent, they'll likely say, "True, and you'll do both."

    Sample Problem

    Let's say x = 4 and y = 17. Is the statement, "x is prime or y is a perfect square" true?

    Again, we'll split the statement into atoms and figure out the truth of each atom. We first ask whether x is prime. Well, 4 has a factor of 2, so this part is definitely false. There's still hope though! Is y a perfect square? Four squared is 16 and five squared is 25, which skips over y. This statement gets a big fat "False" as well. Since neither part is true, we're forced to conclude that the whole statement is false.

    To summarize:

    • Not (or negation): "not A" is true when A is false, and is false when A is true.
    • And (or conjunction): "A and B" is true when both A is true and B is true, and is false when either is false.
    • Or (or disjunction): "A or B" is true when either A or B (or both!) is true, and is false when both are false.

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