Deductive Reasoning

Deductive Reasoning 2-1 Introduction In this chapter we will discus how we prove certain theorems by making use of some undefined terms. The terms point, line and plane are not defined in geometry. By agreement, we have accepted they do exist but cannot be defined. But the idea of a point, line and plane can be represented on a sheet of paper. With the help of these undefined terms, we define the other terms in geometry. We make use of these defined terms to deduce the hypothesis and postulates that are further used to prove the theorems. This is how we go about in making conclusions. 2-2 Deductive reasoning is proof Deductive reasoning is the process of making conclusions from the statements that are true or accepted as true and arrive at true or acceptably true statements. The process comprises of three steps: 1. General statement. 2. Particular statement. 3. Deduction. 1. General statement: The general statement made is property of the whole class (or set) of objects, such as a class of oxen: All oxen have humps (a flashy lump on the back of an animal such as camel). 2. Particular statement. The particular statement relates some members of the class to the general statement. All zebus are oxen. 3. Deduction A logical deduction follows by co-relating the general and particular statement with each other: All zebus have humps. To arrive at a conclusion, we will make use of the transitive property which states that ‘equal things are equal to each other’.

Thus if

A= B and

B=K,

General Statement

Particular Statement

then

A=K

Conclusion

Now, the general statement is ‘all oxen have humps’. By correlating it with the particular statements ‘all zebus are oxen’, we conclude that ‘all zebus have humps’. This can be explained in the language of sets as follows: From the two statements ‘all oxen have humps’ and ‘all zebus are oxen’ it is obvious that the set of ‘humps’ is the universal set, set of oxen is its subset and the set of zebus is the subset of oxen. So, this can be represented in a Venn diagram.

2-3 Observation, measurement and experimentation a. Observation: If we look at two figures given below, we cannot predict which has a longer area.

We are not sure about their areas, but they have equal areas. So we conclude that observation cannot stand as proof. b. Measurement In measurement, a certain amount of uncertainty is always involved. For example, if we say that the height of a person is 183 cm, then it can be 182 cm and 9 mm or 183 cm and 1 mm as the last digit is always considered to be uncertain digit. So, we conclude that measurement does not serve as proof. c. Experiment If we have two red balls and two white balls and all are alike, and we pick one from the bag they are kept in, the probability of a white ball is 2/4 and the probability of a red ball is also 2/4. But out of four draws, it may be 3 red balls and 1 white ball, 2 red balls and 2 white balls. All red balls and no white balls. 1 red ball and 3 white balls. So, we conclude that experimentation cannot serve as proof. 2-4 Deductive reasoning in geometry The systematic combination of, undefined, defined and assumptions leads to logical conclusions and is called deductive reasoning in geometry. It will be clear from the following flow chart.

2-5 Algebraic postulates used in geometry 1. If A=B and B=C then A=C. This is called transitive postulate. 2. An equation can be solved by substituting the values of the variables. Thus, if x − 8 = y − 6 is an equation and we substitute 8 for y, then x −8 = 8− 6 ⇒ x −8 = 2 ⇒ x = 2+8

Transposing 8 to RHS

⇒ x = 10

3. The sum of the parts of a whole is equal to the whole. or

3 4 + =1 7 7

How, here 3 and 4 are the parts of a ‘whole’ 7. By, taking the lowest common multiple (LCM) of the above function we have

3 4 3+ 4 7 + = = = 1 (the whole) 7 7 7 7 4. Identity congruence: Any quantity is always equal to itself. Thus a=a, ara of a circle is equal to itself etc.

5. If an equal quantity is added to two mutually equal quantities then the sums are equal If A=B and P=Q Then A+P = B+Q and A+Q = B+P 6. If an equal quantity is subtracted from two mutually equal quantities, then the difference are equal If A=B and P=Q Then A-P = B-Q and A-Q=B-P 7. If two quantities are equal and each is multiplied by equal quantities, then their products are also equal. If A=B and P=Q Then AP=BQ Thus, if the capacity of a can is 5 litre, then the capacity of 4 such cans will be 20 litre. 8. If two equal quantities are divided by equal quantities separately then their quotient are equal Thus, if A=B and P=Q Then A/P = B/Q, provided that P ≠ 0 and Q ≠ 0. (Note: Division by ‘zero’ is undefined) 9. Two equal numbers when raised to like powers remain equal to each other. If A=B, then ( A) n = ( B ) n 10. Two equal quantities from equal surds with equal roots. If A=B, then ( A) If x 4 = 625, then

1

n

= ( B)

1

n

x = (625)

1 4

1

x = (5 × 5 × 5 × 5) 4 1

x = (54 ) 4 x=5 2-6 Geometric Postulates 1. One and only one line can pass through two different given points.

2. Two different lines intersect at one and only one point.

3. A line segment corresponds to the shortest distance between two points.

4. One and only one circle can be drawn with any given point as centre and a given line segment as its radius.

In the above figure 2-7, the only circle ‘O’ can be drawn with O as centre and OP as its radius.

5. Any geometric figure can be moved in any direction without change in its size or shape.

Thus, rectangle A has been moved to the new position as shown in figure 2-8 6. A line segment has a unique mid-point.

Thus, M is the only mid-point of PQ , as shown in figure 2-9 7. There is one and only one bisector of a given angle.

Thus, PS is the only bisector of ∠RPQ . 8. Through any given point on a line, one and only one perpendicular can be drawn to the line.

Thus, CD ⊥ AB at point P on AB , as shown in figure 2-11 9. Through any point outside a given line, one and only one perpendicular can be drawn to it.

Thus, PQ , is the only perpendicular to AB from the point P outside the AB , as shown in figure 2-12. 2-7 How to determine hypothesis and conclude This is very logical process. It can be explained by taking the statement ‘an agitated cobra attacks’. The other version of the statement is ‘if a cobra is agitated, then it attacks’. Now, the subject of the first statement ‘an agitated cobra attacks’ is ‘an agitated cobra’ and the predicate is ‘attacks’. In this statement the subject ‘an agitated cobra’ is the hypothesis and the predicate ‘attacks’ is the conclusion. In the statement ‘if a cobra is agitated, then it attacks; the subject is ‘if a cobra is agitated’ and the predicate is ‘then it attacks’, the subject that is, ‘if a cobra is agitated’ is the hypothesis and the predicate, that is, ‘then it attacks’, is the conclusion. In these statements, ‘if’ and ‘then’ can be ignored. The hypothesis indicates what is given and the conclusion points to what is to be proved. So ‘if’ statement is the hypothesis and the ‘then’ statement is what we have to prove. 2-8 Converse of a statement The converse of a statement is formed by changing the hypothesis into conclusion and vice-versa. To do so, interchange ‘if’ into ‘then’ and then into ‘if’. For example, in the statement, ‘if a line is parallel to one side of a triangle, then it divides the other two sides proportionally; its converse can be obtained by interchanging the ‘if’ and ‘then’ statements. Hence, its converse becomes, ‘if a line divides two sides of a triangle proportionally, then it is parallel to the third side’.

It should be noted carefully that ‘the converse of a true statement needs not be necessarily true’. Thus, the statement, ‘the adjacent sides of a rectangle are not equal’ is true. But is converse needs not be true. The other point to be noted is that ‘the converse of a definition is always true’. For example, ‘a quadrilateral is a polygon of four sides’ is the definition of a quadrilateral. Its converse, ‘a polygon of four sides is called quadrilateral’, is true. Thus, both the definition and its converse are true. 2-9 How to prove theorems For proving a theorem, we should divide the given statement of the theorem into two parts; hypothesis (if) and conclusion then, as explained in article 2-7. Hypothesis contains what is given to us and conclusion points out to what has to be proved. To begin with, make a neat diagram and label it with helpful symbols for equal sides, equal angles, right angles, question marks for parts to be proved etc. The diagram should be made on the right hand side. Now, divide the space on the left equally for two columns. In the left column write the statements and on the right provide the reasons, such as, already proved theorems, definitions, postulates, etc. It should be noted carefully that all the statements must refer to the diagram on the right side. Use capital letters to name the corners of the diagrams, see the figure to understand the whole procedure. Prove: Write the given statement. (For example, sum of the angles of a triangle is equal to 180o Given (for example, ABC ) To prove: (for example, m∠A + m∠B + m∠C = 180o

Proof: Statement 1. 2. 3.

Reason 1. 2. 3.

-------------------------------------------------