Logic and Proof In logic (and mathematics) one often has to prove the truthness of a statement made. A proposition is a (declarative) sentence that is either true or false. Example: An odd number is prime. Example: A horse has four legs. A proposition is true only if it is always true. A proposition that is sometimes true is a false statement. So the first proposition is a false statement. In logic, we often use the letter p to represent a proposition. A conditional is a statement written in the form of if p, then q format, where p is the hypothesis, and q is the conclusion. Example: An odd number is prime. This statement can be stated in the if-then form as follows: If a number is odd, then it is prime. In the above statement, a number is odd is the hypothesis, and the number is prime is the conclusion. The negation of a proposition p is the proposition not p. Example: The negation of the proposition 21 is divisible by 3 is: 21 is not divisible by 3. The negation of a statement made about the entire collection of a group is a statement that says some object is not in the group. Example: All dogs are carnivorous The negation of the above statement is: Some (one or more) dogs are not carnivorous. Example: All angles are right Negation: Some (one or more) angles are not right. For any proposition Example: p is the statement: x = 3 The negation of p would be: x 3 p, p or its negation is true, and the other false One or the other statement must be true, the other false. One way to prove a statement is to make a (true) statement about a characteristic of all objects in a certain collection, then make a particular statement about one or more of the objects in that collection. It then follows logically that those
particular object(s) must possess the said characteristic. Example: General Statement: All humans are bi-pedal Specific Statement: John is a human Conclusion: John is bi-pedal Example: General Statement: All even numbers are divisible by 2 Specific Statement: all numbers end in 4 are even numbers Conclusion: all numbers end in 4 are divisible by 2 To use the above three steps argument (called a syllogism) to arrive at a conclusion is called deductive reasoning. It ensures the correctness of the conclusion as long as the first and second statements are true. We use deductive reasoning often to prove theorems in geometry. To prove a conditional, one must show that the conclusion of the statement, q, always follows from the hypothesis, p. To prove a statement about a collection of objects, one must show that the statement is true for all conceivable objects in the collection. It is not enough to come up with just one instance of when the statement is true. Example: All odd numbers are prime. To prove this statement, it is not enough to say that 3 is an odd number, and 3 is prime, therefore the statement is true. What you have provided is an example of when the statement is true, but not a prove. To prove the statement, one must prove that every odd number is prime. To disprove a statement about a collection of objects, one only needs to provide one instance of an object in the collection that does not agree with the statement. In other words, one only needs to provide one counter-example to disprove a statement. 15 is an odd number, but 15 is not a prime. Therefore, 15 is a counter-example to the above statement. This disproved the statement All odd numbers are prime. If a statement made is about a finite number of objects, it is possible to prove the statement by examining every object mentioned in the statement, and show that the statement is true for those objects. Example: All odd numbers greater than 2 and less than 8 are prime. We may phase this statement in the if/then form as: If a number is odd and greater than 2 and less than 8, then the number is prime.
Since there are only three numbers that satisfy the hypothesis of the statement, namely 3, 5, and 7, one only needs to prove that the statement is true for these three numbers, then the statement is proved. One can never prove a statement by physical measurements or experiment. For example, if you are asked to prove that an angle is a right angle, you may not use a protractor and measure the angle and say that, according to your measurement, the angle is 90. It is understood that measurements are never exact but are just estimates. The accuracy of the measurement is dependent on the precision of the measuring device, which always have physical limitations in its accuracy. Given a statement in the form if p then q, its converse is the statement if q then p Example: If x = 3, then x 2 = 9 The converse of this statement is the statement: If x 2 = 9, then x = 3. Example: If two angles are complementary, then they are both acute angles The converse of this statement is: If two angles are both acute, then they are complementary. Note that even if a statement is true, its converse is not necessarily true. The contra-positive of a if p then q statement is the statement: if not q, then not p Example: If a number ends in 0, then it is divisible by 10 The contra-positive of this statement is: If a number is not divisible by 10, then it does not end in 0 An if-then statement and its contra-positive is logically equivalent. This means that an if-then statement and its contra-positive is always both true or both false at the same time. Sometimes, to prove a statement, it may be easier to prove its contra-positive. Another method of prove is prove by contradiction. In order to prove that a statement is true, one may first assume that the statement to be proved is false, and proceed to show that this leads to a logical contradiction, hence establishing that the assumption (that the statement is false) is not true, proving the statement. Proof by contradiction is used most often in trying to prove statements that establish the negation of something. For example, to prove that an object does not possess a certain property, or that an object of a certain property does not exist, or that two objects are not equal to each other. Sometimes, to prove statements that are made about the set of natural num-
bers, one may use the principle of mathematical induction. Below is a guideline to use in trying to prove a mathematical statement using a two-column format: Find out what is the hypothesis (the given) and what is the conclusion. Not every mathematical statement is explicitly made in the if-then format, but a mathematical statement that needs to be proved can always be restated in the form of if-then. Make a two column table. On the first column you will write the statements that logically follows the previous statement, starting with the given. On the second column are the reasons that support the statement that you made on the first column. The reasons you may use to support your argument include definitions, other accepted statements (postulates or principles), and other known, alreadyproved, theorems. In a geometric prove, it is always advisible to draw a diagram of the situation, if one is not already provided. Some well-known, accepted principles and theorems from algebra and geometry: From Algebra: For any object x, x = x (an object always equals itself. This is the reflexive property). If a = b, and b = c, then a = c (the transitive property) If a = b, then anywhere a is used in a statement, b can be used instead and the meaning of the statement is unchanged. This is the substitution property. If a = b and c = d, then a + c = b + d (addition postulate) If a = b and c = d, then a c = b d (subtraction postulate) From Geometry: Between any two points there is one and only one (straight) line. (Euclid s first postulate) Any line segment can be extended indefinitely in one or both directions to form a straight line. (Euclid s second postulate) Given a line segment, one can draw a (unique) circle with the given segment as radius and one of the end point of the segment as center (Euclid s third postulate, this can be generalized to the more easier to apply non-collapsing compass version: Given a line segment and a given point, one can draw a circle with its center at the given point and the length of the radius equal to the length of the given line segment.
All right angles are equal to each other (Euclid s fourth postulate). The intersection of two (different) lines is a point. The shortest distance between two points is a (straight) line. The sum of the measure of two adjacent angles is equal to the measure of the angle formed by the non-common sides of the two adjacent angles. (Angle Addition Postulate) In a line segment, if points A, B, C are colinear and point B is between point A and point C, then AB + BC = AC. (Segment Addition Postulate) Using these accepted postulates and known facts from algebra and geometry, together with other theorems and facts, one can prove other statements and theorems. Example: Prove that If two angles are supplementary to the same angle, then they are congruent to each other The hypothesis in this statement is that two angles are supplementary to the same angle, and the conclusion is that then the two angles are congruent to each other. It helps to draw a picture to see what is going on: 1 2 1 3 In the given picture, 2 and 3 are both supplementary to the same 1, we need to prove that 2 = 3 Statements Reasons 1. 2 supplement to 1 3 supplement to 1 1. given 2. m 2 + m 1 = 180 m 3 + m 1 = 180 2. definition of supplementary angles 3. m 2 + m 1 = m 3 + m 1 3. substitution 4. m 2 = m 3 4. subtraction postulate 5. 2 = 3 5. definition of angle congruence Example: Prove that Vertical Angles are Congruent In order to prove this statement, one must prove that whenever two lines intersect and form a pair of vertical angles, then the angle pairs must be congruent. The hypothesis in this statement is two (arbitrary) lines intersect to form a pair of vertical line, and the conclusion is the pair of vertical lines are congruent. We should draw a diagram to illustrate what we wanted to prove:
B E A C D In the picture above, AC intersects BD at point E to form vertical angles pair AEB and DEC. We must prove that AEB = DEC. Proof: Statements Reasons 1. AC intersects BD at point E 1. given 2. AEB and DEC are vertical angles 2. definition of vertical angles 3. m AEB + m DEA = 180 m DEC + m DEA = 180 3. angle addition postulate 4. AEB and DEA are supplementary DEC and DEA are supplementary 4. definition of supplementary angles 5. AEB = DEC 5. Angles supp. to same are = Notice that we used a theorem we just proved to prove another theorem.