 In this presentation we consider the role of proofs in mathematics and we look into the general types of mathematical proof as well as the manner by which these are constructed. Proof is central to the discipline of mathematics and the practice of mathematicians. It is essential in doing communicating and recording mathematics. In fact, it is facilitated the development and delivery of new theories, concepts and methods in mathematics. The strength and stability of mathematics as well as its reliability as a tool and method for study and understanding real world phenomena rests in the validation provided by the mathematical proof with its rigor, precision as well as its appealing creativity. What is mathematical proof? A mathematical proof consists of a hypothesis carefully stated assumptions using precise language and definitions, arguments using logical reasoning aided by previously established and assumed attributes of the mathematical concepts and quantities involved so as to obtain a valid conclusion. All areas of mathematics involve structures such as the real numbers and operations on these as well as assumed properties which form an axiomatic system. Mathematical statements regarding properties of these elements are established using rules of logic. Individuals studying mathematics use logical reasoning to synthesize or decide on the validity of claims or proofs, arguments. The application of rules of logic is used to explain and justify why mathematical statements or conclusions are correct. Of what use are proofs in mathematics? Mathematicians and students of math use proofs for the following purposes. One, to verify that a statement is true. Two, to explain why a statement is true. To communicate mathematical knowledge. Four, to discover or create new mathematics. And five, to systematize statements into an axiomatic system. Here are some examples of mathematical statements that are proven. One, the sum of the angles of a triangle is 180 degrees. Two, the Pythagorean theorem. Given a right triangle, the sum of the squares of the respective lengths of the sides perpendicular to each other is the square of the length of the hypotenuse. Three, the sum of the first n natural numbers is the product of n by n by 1 divided by 2. Four, the solution to the quadratic equation is given by the quadratic formula. Five, if x is a number greater than 1, then x plus its multiplicative inverse is greater than or equal to 2. Six, the square of a two-digit number ending in 5, for instance 75, is found by multiplying the tenth digit by its successor integer and multiplying this by 100, finally adding 25. And seven, the sum of the first n odd integers is n squared. The following are forms of statements proven in mathematics. One, p equals q, that is something equals something else. For instance, we say the sum of the first n natural numbers equals n times n plus 1 divided by 2. Two, p implies q. If p is true, then q is true. For example, if ax squared plus bx plus c equals 0, then x is negative b plus or minus the square root of b squared minus 4ac divided by 2a. Or if set a is a subset of set b, then a union b equals b. Three, p if and only if q. If p is true, then q is true. And conversely, if q is true, then p is true. For example, if a is a subset of b, this is true if and only if the intersection of a with b equals a. Four, p has a specified property or it has some other interesting and relevant property. For example, the collection or set of all prime numbers is an infinite set. Five, for all x, p of x is true. That is, all entities of a certain kind behave in a certain way. For example, all odd integers have squares which are odd integers. Number six, there exists x such that p of x is true. That is, there is an entity that behaves in a certain way, p of x. For example, for non-zero real numbers a and b, there is a non-zero real number that is the multiplicative inverse of the product of a with b. In constructing proofs, mathematicians pay attention to the transition from one statement or result to a subsequent statement and that this is valid and done according to rules of logic. Oftentimes, distinctions are made between deductive reasoning and inductive reasoning. Deductive reasoning involves chains of statements that are logically connected. Developed by ancient Greeks, the procedure and reasoning characterized early mathematical thinking and logical thinking in other domains. Inductive reasoning is used when generalizing observations or results from a few cases. This is sometimes used when working with patterns, for instance, for young students or in experimental mathematics. Some types of mathematical proof are the following. One, direct proof. Two, proof by exhaustion. Three, proof by contradiction. Four, existence proof. And five, proof by mathematical induction. The direct proof consists of a chain of statements, each one following logically from the previous one. For instance, to prove the additive cancellation law, a plus c equals b plus c implies a equals b for all real numbers a, b and c. We have first a plus c equals b plus c. This is given then a plus c, the quantity plus negative c equals the quantity b plus c plus negative c. This is justified by the addition property for equality. And then a plus the quantity c plus negative c equals b plus the quantity c plus negative c by associativity of addition. Finally, a plus 0 equals b plus 0 because of the additive inverse property and so a equals b by the identity for addition. A proof by exhaustion relies on checking all cases. In some fields of mathematics such as number theory, it is a common practice to use categories that exhaust all possible cases to construct a proof. For example, a computer or calculator program or a spreadsheet may be written to prove that there are only two three digit numbers with a property that the numbers themselves are the sum of the cubes of their digits. These are 153, which is equal to 1 cube plus 5 cube plus 3 cube and 407, which is 4 cube plus 0 cube plus 7 raised to the third power. Another example to show that the absolute value of x is less than a for a given positive number a, if and only if minus a is less than x and this is less than a, one can consider two cases namely x greater than or equal to 0 and 2 x less than 0. In case one, the absolute value of x equals x and so the assumption yields x less than a. For case two, the absolute value of x equals negative x, hence the absolute value of x less than a yields minus x less than a and multiplying both sides by negative 1, we reverse the direction of the inequality. This is a property for numbers that may be proven separately and we have minus a less than x. Considering both cases, whether x is negative or not, we get minus a is less than x and this is less than a. The basic idea for the proof by contradiction is the principle of excluded middle p or not p, there is no middle ground. So, not p yields a contradiction therefore, one must conclude p. Suppose we are trying to prove some statement capital P, we take its negation or its opposite to be true and if this yields a contradiction to what is in the hypothesis or what has been previously established mathematically, then we must conclude p. Some examples, suppose for two sets A and B, the union of A with B equals A, then it must be the case that B is a subset of A. The proof consists of supposing on the contrary that B is not a subset of A. Then this implies there is an element small b of set capital B that is not in the set capital A, but A union B contains all elements that are in either one of the sets A or B. With the element small b in the set capital B and not in A and necessarily small b is in A union B, we have a contradiction to the assumption that A union B equals A, since A union B contains the element B which is not in A. Thus assuming the negation of the conclusion B is a subset of A, this is inconsistent with the condition attributed to the set A and so we must reject this and hence conclude the opposite that is B must be a subset of A. Other examples which you might consider one n squared is even if and only if n is even, one might assume the contrary and arrive at the contradiction or the square root of 2 is irrational. The proof consists of assuming that the square root of 2 is rational and one arrives at the contradiction or there are an infinite number of primes. The proof for this consists of starting with the assumption that there is a finite number of primes and one ends up with a contradiction. Finally, if a non-negative real number x can be made less than any positive real number then x is equal to 0. This again may be proven by contradiction. An exercise that might be considered also, if the square of a number is a multiple of 3 then the number is a multiple of 3. Now, a slightly different kind of proof by contradiction is that of the counter example. Here is our counter example. Euler conjectured that there do not exist integers x, y, z and w such that x to the fourth power plus y to the fourth power plus z to the fourth power equals w to the fourth power. 200 years later, Neum Elks of Harvard showed that there indeed are such numbers namely 2,682,440 raised to the fourth power plus 15,365,639 raised to the fourth power plus 18,796,760 raised to the fourth power equals the fourth power of 20,615,673. The existence proof. Proofs to establish existence of a particular mathematical entity or quantity or solution to some equation are used widely in advanced mathematics. For example, the statement that an nth degree polynomial equation with real number or complex number coefficients has roots numbering in and this is proven using this type of proof. However, the machinery required to establish this theorem known as the fundamental theorem of algebra is deep mathematical theory using advanced topics in the area of abstract algebra or complex analysis. To illustrate this theorem, one can cite the quadratic equation A x squared plus B x plus C equals 0, a second degree polynomial equation with two solutions given by the quadratic formula x equals the quantity negative B plus or minus the square root of B squared minus 4ac all divided by 2a. Finally, for the inductive proof one can use the image of an infinite staircase through which one can attain any particular step if we can establish two things. One, that getting to the first step is verifiable or attainable and two, that from a prior step the next step is also verifiable. So, to prove that a statement is true for all natural numbers n, first prove that it is true for n equals 1. This is the base case or the basis step and then show that if it is true for n then it is also true for n plus 1. This is called the inductive step. The base case need not be 1, it can be 2, 3, etcetera. If the base case is k, then the statement is to be proven for n greater than or equal to a. Some examples and one, the sum of the first n positive integers is equal to n times n plus 1 divided by 2. For the proof, first if n equals 1, obviously the sum of 1 equals 1 times 1 plus 1 divided by 2. Next, if the sum of the first j integers from 1 to k equals k times k plus 1 divided by 2, then we show that the sum of the first k plus 1 positive integers is k plus 1 times k plus 1 plus 1 divided by 2. Indeed, if we take the sum of the first k plus 1 integers, this is equal to the sum of the first k integers plus k plus 1. And we expand this into k times k plus 1 divided by 2 plus k plus 1 and this is equal to factoring k times k plus 1 divided by 2 plus 2 times k plus 1 divided by 2 simplified finally, into k plus 1 times k plus 1 plus 1 divided by 2. The steps for the proof by induction are the following. One, state the hypothesis or claim clearly, this is p of n. Two, the basis step establish that p of k is true, k is usually 1 and the inductive step assume that p of n is valid and prove that p of n plus 1 follows that is p of n implies p of n plus 1. Some exercises that may be done the same formula, the sum of the first squares j squared j from 1 to n equals n times n plus 1 times 2 n plus 1 divided by 6 or prove that a minus b is a factor of a to the n minus b to the n for every positive integer n. And we have a hint regarding a to the k minus 1 minus b to the k minus 1. Finally, prove that a plus b is a factor of a to the 2 n plus 1 plus b to the 2 n plus 1 for every positive integer n. There is a variant of the proof by induction and this is strong induction. In some instances when proving p of n plus 1, we want to be able to use p of j for j less than or equal to n instead of just p of n. This is sometimes referred to as strong induction. The basis step is the same. The inductive step is modified slightly that is p of k, p of k plus 1 up to p of n holds and then one shows that p of n plus 1 is true. For example, any item costing n greater than 7 copics can be bought using only 3 and 5 copic coins. The proof for the basis step we verify for k equals 8. Then assume that it is true for k equals 8, 9 up to n. We then show that the claim is true for n plus 1. Note that if k equals 9, then 9 is 3 plus 3 plus 3 and 10 equals 5 plus 5. Now, n plus 1 minus 3 is n minus 2 greater than or equal to 8. Hence, p of n minus 2 is true. Further, n plus 1 equals n minus 2 plus 3. Hence, the statement is true for n plus 1 assuming it is true for k equals 8, 9, 10 up to n. Some notes on the proof by contradiction. Contradiction is useful in proving statements in the form. It is not true that star holds where star is a mathematical statement. If we are proving that something is not true, proof by contradiction may be useful. Proof by contradiction adds an extra hypothesis that the purported conclusion is false. Most propositions in mathematics involve setting some hypothesis and using rules and known results to arrive at a conclusion under the conditions set by the hypothesis. A proof by contradiction uses this hypothesis plus the negation of the conclusion to establish an untenable conclusion or what we call a contradiction. Some examples 1, 0 does not have a multiplicative inverse or 2, in calculus, limits of functions if they exist are unique. To conclude, here are some famous results or conjectures that have not found their proofs. And so, this is a challenge to succeeding mathematicians for maths last theorem. This states that there are no integers x, y, z such that n is greater than 2 and x to the n plus y to the n equals z to the n. A second unproven conjecture that twin primes conjecture there are an infinite number of twin primes, primes that differ by 2 for instance 11 and 13, 17 and 19 or gold box conjecture every positive integer can be expressed as the sum of 2 prime numbers. The discussion and examples given in this presentation show that mathematical proof being characterized by rigor and precision is the scaffolding for the structures and concepts of mathematics. Much creativity is employed and nurtured in the process of constructing mathematical proof. In many ways, because of the training in mathematical proof, students develop a critical eye to guard against invalid statements or indeed versions of fake news.