If you are in class 11 and have opted for Maths as one of your core subjects then you must be feeling the hike in difficulty in comparison with older classes. Introduction to different branches of Mathematics like calculus, number theory, vectors, numerical analysis and algebra occurs in this particular class only. Mathematical Reasoning class 11 in one such topic which concentrates on the theoretical part of the Maths. Read on to know intricate details around this chapter!
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Introduction to Mathematical Reasoning Class 11
The principle of Mathematical Reasoning is a branch of maths where you verify the truth values of the provided statements. Logic is the topic, which deals with the reasoning methods. It offers rules to prove the theorem by finding out the validity of a known argument. Let us know what reasoning, the concepts in mathematics are and be familiar with how to resolve questions easily.
The statement is the basic unit of Mathematical Reasoning. A statement is an assured sentence that each sentence is true or false but cannot be both. If a statement is true then it is called a valid statement or else, it is called an invalid statement. Statements are represented by small letters like p, q, r and more.
Example: ‘Addition of two prime numbers is even.’
The provided statement can be either true or false because the addition of two prime numbers can be an even or odd number. The reasoning for this statement is not acceptable mathematically, as the sentence is uncertain. So this sentence is accepted only when it is changed to ‘Either true or false, but cannot be both at the same instance.’ So, the basic unit required for Mathematical Reasoning is called a statement and hence the definition.
Types of Reasoning
Reasoning can be of 2 types in mathematical terms.
- Inductive Reasoning
- Deductive Reasoning
The other types of reasoning include abductive induction, backwards induction, critical thinking, counterfactual thinking and intuition. These 7 types of reasoning are mainly used to conclude. In this article, we have discussed the two main types – i.e., inductive and deductive reasoning.
Inductive Reasoning is a method used to validate the statement that is verified with a definite set of rules after that to generalise. Inductive reasoning concept is employed for the principle of mathematical induction and is not considered in geometrical proofs, as it is generalised. Follow the example shown below to understand the inductive reasoning better.
Statement: The cost of supplies is Rs 100 and the labour cost to manufacture the item is Rs 50. The selling price of the item is Rs 500.
Reasoning: It can be said that from the given statement, the item will offer a good profit by selling it.
Deductive Reasoning principle is exactly contradictory to that of the principle of induction. In deductive reasoning, based on the previous facts for a given statement, you will make a conclusion. The given example shown below will assist in knowing deductive reasoning concepts better.
Statement: Pythagoras Theorem holds good for right-angled triangles.
Reasoning: If triangle ABC is a right-angled triangle, then it follows Pythagoras Theorem.
Types of Reasoning Statements
The 3 main types of Reasoning Statements are
- Simple Statements
- Compound Statements
- If-then Statements
Also Read: 3D Geometry Class 11 Notes
Simple Statements -Simple statements are direct sentences that don’t need much reasoning and are simple to solve. These statements are declarative enough to decide true or false.
Statement: The sun sets in the west.
In the above statement, there is no modifier, and hence it can be concluded as true.
Compound Statements – Compound statements are made up of combining two or more statements with the help of definite connectives. The connectives used to club statements are ‘and’, ‘or’ and more. With the aid of these statements, the mathematical deduction concept can be executed easily. Follow the example for better understanding.
Statement 1: Square of 7 is 49
Statement 2: 7 is also an odd number
Compound Statement: Square of 7 is 49, and 7 is also an odd number.
Compound Statement: A rectangle has four sides, and the sum of the interior angles of a rectangle is 360°.
Statement 1: A rectangle has four sides
Statement 2: The sum of the interior angles of a rectangle is 360°
From the above two examples ‘and’ is used as connective to form the statements.
If-then Statements – Here we come across if-then statement combinations. For example, if ‘A’ is true, then ‘B’ can be verified to be true or if you prove that ‘B’ is false, then ‘A’ is also false.
A: 6 is multiple of 36
B: 6 is a factor of 36
Given that one of the statements that are ‘A’ is true; thus A or B is true.
Method of Deducing Mathematical Statements
For making significant deductions from the known statements, three methods are generally used.
- The negation of the Certain Statement
- Contradiction Method
- Counter Statements
The negation of the Certain Statement – In this method, you make new statements from the previous ones by denying the given statement. Particularly here you deny the provided statement and express as a fresh statement. Go through the below-shown example to understand the concept.
Statement 1: ‘Sum of cubes of three natural numbers is positive.’
Now if you negate the above statement, then it will be.
Statement 2: ‘Sum of cubes of three natural numbers is not positive.’
From the above example, “not” is used to reject the statement provided. For the conclusion that if Statement 1 is a mathematically acceptable statement, then the rejection of Statement 1 (Represented by Statement 2) is also a statement.
Contradiction Method – In this method, we believe that the known statement is false and then attempt to prove the statement is wrong.
Consider, for all integers q, if q2 is odd, q is odd.
Now you have to take the rejection of the given statement and deduce it to be true.
So consider q2 is even. So the conclusion that p is even, then q2 that is the product of q is also even. This statement will disagree with the possibility that q2 is odd. Therefore, the proposition is true, and the possibility is false.
Counter Statements – The counter-statement is a method to prove that in which areas the provided statement is not applicable.
A: If P is a prime number, then P is always even.
Now, you have to come across a negative statement to make the above statement false. You know that 3 is the prime and also an odd number that is divisible by itself and 1. Hence, you can say that 3 is an odd prime number. Therefore, the given statement ‘A’ is not true for all the prime numbers and is not valid.
This was all in Mathematical Reasoning class 11. We hope this blog helped you understand the topic better. If you are confused regarding what career pathway to choose after completing your school education, then Leverage Edu experts are here to help you out. Stay tuned with us for more study notes and career guides!