Difference between inductive and deductive reasoning :
Inductive and deductive reasoning are often confused. This section introduces the concept of reasoning and gives you tips and tricks to keeping inductive and deductive reasoning straight.
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
Apart from the above two examples, let us look at the basic stuff of inductive reasoning and deductive reasoning separately to understand the difference between them better.
Looking for patterns and making conjectures is part of a process is called inductive reasoning.
It consists of three stages.
(i) Look for a pattern.
Look several examples. Use diagrams and tables to help to discover a pattern.
(ii) Make a conjecture.
Use the examples to make a general conjecture. A conjecture is an unproven statement that is based on observations. Discuss the conjecture with others. Modify the conjecture, if necessary.
(iii) Verify the conjecture.
Use logical reasoning to verify that the conjecture is true in all cases.
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.
(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.
Example 1 :
Sketch the next figure in the pattern.
Solution :
Each figure in the pattern looks like the previous figure with another row of squares added to the bottom. Each figure looks like a stairway.
So, the sixth figure in the pattern must have six squares in the bottom row.
Example 2 :
Describe a pattern in the sequence of numbers. Predict the next number.
1, 4, 16, 64, ............
Solution :
Each number is four times the previous number.
So, the next number is 256.
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 :
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 3 :
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.)
After having gone through the stuff given above, we hope that the students would have understood "Difference between inductive and deductive reasoning".
Apart from the stuff given above, if you want to know more about "Difference between inductive and deductive reasoning", please click here
Apart from the stuff given on "Difference between inductive and deductive reasoning", if you need any other stuff in math, please use our google custom search here.
WORD PROBLEMS
HCF and LCM word problems
Word problems on simple equations
Word problems on linear equations
Word problems on quadratic equations
Area and perimeter word problems
Word problems on direct variation and inverse variation
Word problems on comparing rates
Converting customary units word problems
Converting metric units word problems
Word problems on simple interest
Word problems on compound interest
Word problems on types of angles
Complementary and supplementary angles word problems
Markup and markdown word problems
Word problems on mixed fractrions
One step equation word problems
Linear inequalities word problems
Ratio and proportion word problems
Word problems on sets and venn diagrams
Pythagorean theorem word problems
Percent of a number word problems
Word problems on constant speed
Word problems on average speed
Word problems on sum of the angles of a triangle is 180 degree
OTHER TOPICS
Time, speed and distance shortcuts
Ratio and proportion shortcuts
Domain and range of rational functions
Domain and range of rational functions with holes
Graphing rational functions with holes
Converting repeating decimals in to fractions
Decimal representation of rational numbers
Finding square root using long division
L.C.M method to solve time and work problems
Translating the word problems in to algebraic expressions
Remainder when 2 power 256 is divided by 17
Remainder when 17 power 23 is divided by 16
Sum of all three digit numbers divisible by 6
Sum of all three digit numbers divisible by 7
Sum of all three digit numbers divisible by 8
Sum of all three digit numbers formed using 1, 3, 4
Sum of all three four digit numbers formed with non zero digits