Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R Assertion A: Let $R \subseteq A \times A$ be a binary relation as defined below: $A = \mathbb{N} \times \mathbb{N}$; $((a,b), (c,d)) \in R$ if only if $a \le c$ or $b \le d$. This binary relation R is a partial order. Reason R: A binary relation will be a partial order if this satisfy reflexive, anti-symmetric and transitive property. In the light of the above statements, choose the most appropriate answer from the options given below
To determine the correctness of Assertion A and Reason R, let's analyze each statement step-by-step:
- Assertion A: It states that the binary relation \( R \), defined on the set \( A = \mathbb{N} \times \mathbb{N} \), satisfying \(((a,b), (c,d)) \in R \) if and only if \( a \le c \) or \( b \le d \), is a partial order.
- Reflexive Property: For any \( (a, b) \in \mathbb{N} \times \mathbb{N} \), \( (a, b) \le (a, b) \) holds since \( a \le a \) or \( b \le b \) is always true. Hence, \( R \) is reflexive.
- Antisymmetric Property: For \( ((a, b), (c, d)) \in R \) and \( ((c, d), (a, b)) \in R \) implies \( a = c \) and \( b = d \) is not necessarily true, because it is possible for one to be less and the other to be greater (e.g., \( a \le c \) and \( d \le b \) doesn't imply \( a = c \) and \( b = d \)). Hence, \( R \) is not antisymmetric.
- Transitive Property: If \(((a, b), (c, d)) \in R\) and \(((c, d), (e, f)) \in R\), it needs to imply \(((a, b), (e, f)) \in R\). However, it doesn't hold for general ordered pairs. For example, if \( a \le c \) and \( d \le f \), it doesn't necessarily ensure \( a \le e \) or \( b \le f \). Hence, \( R \) might not be transitive.
- Reason R: It correctly states that for a binary relation to be a partial order, it must satisfy reflexive, antisymmetric, and transitive properties.
This statement is indeed true for defining a partial order relation on any set.
The most appropriate answer is: A is not correct but R is correct.
Important Questions from Assertions and Reasons
In this question, two statements are given – one labeled as Assertion (A) and the other labeled as Reason (R) respectively. Assuming that all the information given in the assertion is true, analyze both the reasons along with it and decide which of them is/are correct and whether R is the correct description of A.
Assertion: Oral polio vaccine should be given to all children below the age of 5 years; But children over the age of 5 do not need to take the dose.
Reason: In India, almost all complications of poliomyelitis are found in children below the age of 5.
For the Assertion (A) and Reason (R) given below, choose the correct option from the following :
Assertion (A): Buddhism considers suicide as the best way to attain salvation.
Reason (R): Buddhism claims that the world is full of sorrow and pain.
In the question given below, two statements are marked as Assertion (A) and Reason (R). Mark your answer according to the code given in the alternatives.
Assertion (A) : There has been a huge loss of property in Japan last year.
Reason (R) : Japan was hit by Tsunami last year.
In the question given below, there are two statements marked as Assertion (A) and Reason (R). Mark your answer as per the codes in options.
Assertion (A): Iron rusts when exposed to water.
Reason (R): Iron reacts with the oxygen molecules present in water.
An assertion (A) and a reason (R) are given below.
Assertion (A): Forest cover in the country has gradually decreased.
Reason (R): Encroachment by humans is one of the concerns for the forest department.
Choose the correct option.
A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true, but R is not the correct explanation of A
C. A is true, but R is false
D. A is false, but R is true
