Quantifiers: Expressing and Proving Mathematical Facts

Quantifiers Quantifiers are used in mathematics to express and prove facts. The two quantifiers are: There exists, and for all. These are all we need to look at because of the special nature of mathematical truths. When mathematics is viewed as a subject in its own right as opposed to being a set of tools used in other disciplines, the call of the subject is the theorem, and most mathematical theorems will have one of two forms: there is an object x having property p, for all objects x property p holds. For instance, the equation x2 + 2x + 1 = 0 has a real root. 2 𝑥 + 2𝑥 + 1 = 0 To emphasize the existing statement, we can write it in the following form. The number x, which is real, has the equation x2 + 2x + 1 = 0. 2 𝑥 + 2𝑥 + 1 = 0 Let x be a real number. Since there exists some number x such that x squared + 2x + 1 = 0, then x should equal the square root of 0. 2 𝑥 + 2𝑥 + 1 = 0 The symbol mathematicians use for existence is a back-to-front E, so for example the following statement can be written as follows: there exists an x such that x2 + 2x + 1 = 0. [ 2 ] ∃𝑥 𝑥 + 2𝑥 + 1 = 0 The symbol is known as the existential quantifier. ∃ − 𝑒𝑥𝑖𝑠𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝑞𝑢𝑎𝑛𝑡𝑖𝑓𝑖𝑒𝑟 A way to prove an existential statement of this form is to provide an example of a value "x" that satisfies the property. For example, if we want to prove that there is an integer x such that x squared + 2x + 1 = -1, we can find such an x by taking x = -1. Then x squared + 2x + 1 = (-1)squared which is 1+2(-1)+1=0, which solves this. In some cases, we can prove that a statement is true without actually finding an object that satisfies the property. 2 𝑥 + 2𝑥 + 1 = 1 − 2 + 1 = 0 For example, let`s prove that x^3 + 3x + 1 = 0 has at least one real solution. [ 3 ] ∃𝑥 𝑥 + 3𝑥 + 1 = 0 While we will not solve this cubic equation by finding an actual x, wwe can show that there is a solution by analyzing the function y = x cubed + 3x + 1. 3 𝑦 = 𝑥 + 3𝑥 + 1 A continuous function is one for which the graph is a smooth curve without breaks or jumps. A cubic equation has a nice smooth curve that looks something like this, in many cases. The point is it doesn't have any jumps. If x = -1, this curve has value or this function has value: -1 cubed, which is -1 + 3 x – 1, which is -3 + 1, which is -3. 𝑦 =− 1 − 3 + 1 =− 3 If x equals +1 and y equals 1+3+1, the equation for the curve has the value 5. 𝑦=1 +3 + 1 =5 If x is equal to +1, the curve or the function has value y = 1 + 3 + 1, which equals 5. So the curve lies below the x axis for x values less than -1 and above it for x values greater than or equal to +1. It will cross the x axis at some point; it doesn't matter where. The important point is where the curve crosses the axis. When it does, the y value is 0. 3 𝑥 + 3𝑥 + 1 = 0 We have not stated which x satisfies the equation, nor have we even stated whether x is negative or positive. But we have shown that there exists an x that satisfies the equation. This is an example of an indirect proof, since we have not proven it directly by finding a value for x that satisfies the equation; we simply showed that there is a value of x that satisfies it. If it is 1, then x=0, which makes x cubed + 3x + 1 = 0. 1. [ 3 ] (∃𝑥 < 0) 𝑥 + 3𝑥 + 1 = 0 Or is it 2? There’s an x greater than 0, so it should be x cubed + 3x + 1 = 0? 2. [ 3 ] (∃𝑥 > 0) 𝑥 + 3𝑥 + 1 = 0 In the argument, we looked at the curve at x=-1 and x=+1. We found that it looked something like this: If the curve is between -1 and +1, then it must cross the x axis at some point between those values. However, it does not have to pass through -1 or +1; instead, we can imagine it passing through the origin itself. As long as x = 0 when y = 1, then 3x + 1 = 0. 3 𝑖𝑓 𝑥 = 0 , 𝑡ℎ𝑒𝑛 𝑦 = 𝑥 + 3𝑥 + 1 = 1 The negative sign of 0 means that it is negative at -1. It is positive at 0, so it is somewhere between -1 and 0 on the axis. We know that one of these two statements must be true on the basis of the previous argument. If we can eliminate statement 2 as we have just done, then we can conclude statement 1. So either way, either by refining the previous arguments or by deducing it from previous results, this observation leads us to show that the equation has a root somewhere between -1 and 0. But there is another solution. The equation the square root of 2 = p/q is a false statement, but it can be rewritten as an existence statement: There exist natural numbers p and q such that the equation holds true. 2 = 𝑝∕𝑞 Let exist p, q such that the square root of 2 is p/q. ∃𝑝 ∃𝑞[ 2 = 𝑝∕𝑞] When we prove that the square root of two is not rational, we will do it by showing that there are no values of p and q that satisfy this equation—so we will prove a non-existence statement. (∃𝑝∈𝑁) (∃𝑞∈𝑁)[ 2 = 𝑝∕𝑞] The symbol often used to represent a set of natural numbers is an N with a double line for the diagonal. If we denote the square root of 2 by p/q, then there exists a number p such that q is equal to the square root of 2 over p. (∃𝑝, 𝑞∈𝑁) [ 2 = 𝑝∕𝑞] We will study examples of quantifiers in combination, which can lead to confusion. But when we understand the distinctions between them, we will be able to use them more effectively. So, we're beginning to look at the concept of quantifiers. For example, see if you can prove that the square root of 2 is not rational. That is, prove that there do not exist p and q in the natural numbers such that the square root of 2 equals p over q or that 2 equals p squared over q squared. The following is a proof that the square root of 2 is irrational. [ 2 (∃𝑝∈𝑁) (∃𝑞∈𝑁) 2 = 𝑝 ∕𝑞 2 ] The symbol here is an upside down A, which stands for all. For example, to say that the square of any real number x is greater than or equal to 0, we could write this equation. 2 ∀𝑥(𝑥 ≥0) For all x, x squared is greater than or equal to 0; in other words, for all x, x squared = 2 0.∃𝑥(𝑥 = 0) The same thing happened with a word's use in mathematics. There are different words we can use to express an existence assertion, and there are different words we can use to express a flawless assertion. The universal quantifier tells us that for all real numbers x, x squared is greater than or equal to 0. 2 (∀𝑥 ϵ 𝑅)(𝑥 ≥0) Many statements in mathematics involve two or more quantifiers combined. For example, if we want to say there is no largest natural number, we write "for all m in the set of natural numbers, there is an N in the set of natural numbers such that n is bigger than m." (∀𝑚 ∈𝑁)(∃𝑛 ∈𝑁) (𝑛 > 𝑚) For every natural number m, there is a natural number N such that n is bigger than m. That clearly says there is no largest natural number. (∃𝑛∈𝑁)(∀𝑚∈𝑁)(𝑛 > 𝑚) The result states that there is a natural number n such that for all natural numbers m, n is greater than or equal to n. In other words, this statement says that there is a natural number bigger than all natural numbers which is false. In their fact sheet they wrote, one American dies of melanoma almost every hour. Using our quantifiers, we could write that like this. There exists an American, such that for every hour A dies in hour H. ∃𝐴∀𝐻 [𝐴 𝑑𝑖𝑒𝑠 𝑖𝑛 ℎ𝑜𝑢𝑟 𝐻] But we use mathematical notation in complicated situations and to talk about things we don't yet understand. That means we can't disambiguate; it's really important that we say things in the right order. Where the literal meaning of this statement is captured by number 1. It says, there is a license for which there are two distinct states that that license comes from. But it's false. Licenses are issued by states, you can't get a license that's issued by two separate states. So the correct answer to the question is number 1. The second sentence says that there are two different licenses issued by one state. This is possible, but unlikely. Let's look at number 3. That says, there are 2 licenses and 2 states, different states, different licenses. This is the sentence that the driver's license application meant to say. The sentence means that number 1 is false, but this is not the literal meaning of the sentence. Number 2 is true, but it's not what the sentence is talking about. Number 3 is absolutely clear that the intended meaning is that we do not understand literal meaning number 1—that would be incorrect—but we do understand what the sentence means when we know about driver licenses and state requirements. The correct answer is number 3. Here's why: The first one says, if there is a license and if it were issued by two different states, then that license would be valid in both states. So, it says that if a license is issued by one state and carried into another state, then it is considered valid. The next sentence contains false information—a license may be valid in one state and not in another. The second sentence explains the meaning of an invalid license. Only statement 3 captures the essence of the sentence. Statement 1 is irrelevant, and statement 2 misrepresents the meaning of the sentence by suggesting that some licenses are invalid in some states.