DEDUCTIVE REASONING IN GEOMETRY

Deductive reasoning, unlike inductive reasoning, is a valid form of proof. It is, in fact, the way in which geometric proofs are written. Deductive reasoning is the process by which a person makes conclusions based on previously known facts.

Deductive reasoning uses facts, definitions, and accepted properties in a logical order to write a logical statement.

Conditional statements can be written using symbolic notation, where p represents the hypothesis, q represents the conclusion, and ̆ is read as “implies.”

Here are some examples.

This conditional statement can be written symbolically as follows :

If p, then q or p -> q

To form the converse of an “If p, then q” statement, simply switch p and q.

The converse can be written symbolically as follows :

If q, then p or q -> p

A biconditional statement can be written using symbolic notation as follows :

If p, then q and if q, then p or p <-> q

Most often a biconditional statement is written in this form :

p if and only if q.

Difference between Inductive and Deductive Reasoning

The following examples show how inductive and deductive reasoning differ.

(i) Lily knows that John is a sophomore and Michael is a junior. All the other juniors that Lily knows are older than John. Therefore, Lily reasons inductively that Michael is older than John based on past observations.

(ii) Lily knows that Michael is older than Chan. She also knows that Chan is older than John. Lily reasons deductively that Michael is older than John based on accepted statements

Two Laws of Deductive Reasoning

(i) Law of detachment

(ii) Law of syllogism

Law of Detachment :

If p -> q is a true conditional statement and p is true, then q is true.

Law of Syllogism :

If p -> q and q -> p are true conditional statements, p->q is true.

Examples

Example 1 :

Let p be "the value of x is -5" and let q be "the absolute value of x is 5".

(i) Write p -> q in words.

(ii) Write q -> p in words.

(iii) Decide whether the biconditional statement p <-> q is true.

Solution :

(i) If the value of x is -5, then the absolute value of x is 5.

(ii) If the absolute value of x is 5, then the value of x is -5.

(iii) The conditional statement in part (a) is true, but its converse in part (b) is false. So, the biconditional statement p <-> q is false.

Example 2 :

Write the following statements using symbols.

(i) ∠A measures 30°

(ii) ∠A does not measure 30°

Solution :

Statement

∠A measures 30°

Symbol

Negation

∠A does not measure 30°

Symbol

∼ p

Example 3 :

Let p be "it is raining" and let q be "the cricket match is cancelled".

(i) Write the contrapositive of p -> q.

(ii) Write the inverse of p -> q.

Solution :

(i) Contrapositive : ∼ q -> ∼ p

If the the cricket match is not cancelled, then it is not raining.

(i) Inverse : ∼ p -> ∼ q

If it is not raining, then the cricket match is not cancelled.

Example 4 :

State whether the argument is valid.

Michael knows that if he misses the practice the day before a game, then he will not be a starting player in the game. Michael misses practice on Tuesday so he concludes that he will not be able to start in the game on Wednesday.

Solution :

This logical argument is a valid use of the Law of Detachment. It is given that both a statement (p -> q) and its hypothesis (p) are true. So it is valid for Michael to conclude that the conclusion is true.

Example 5 :

Write some conditional statements that can be made from the following true statements using the Law of Syllogism.

1. If a bird is the fastest bird on land, then it is the largest of all birds.

2. If a bird is the largest of all birds, then it is an ostrich.

3. If a bird is a bee hummingbird, then it is the smallest of all birds.

4. If a bird is the largest of all birds, then it is flightless.

5. If a bird is the smallest bird, then it has a nest the size of a walnut half-shell.

Solution :

Here are the conditional statements that use the Law of Syllogism.

a. If a bird is the fastest bird on land, then it is an ostrich. (Use 1 and 2.)

b. If a bird is a bee hummingbird, then it has a nest the size of a walnut half-shell. (Use 3 and 5.)

c. If a bird is the fastest bird on land, then it is flightless. (Use 1 and 4.)