Let A be the set of all students of a boys school. Show that the relation R in A given by R={(a,b):a is sister of b} is the empty relation and R′={(a,b): the difference between heights of a and b is less than 3 meters } is the universal relation.
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Step-by-Step Solution
Step 1: Analyze the set A
The problem defines set A as the set of all students in a boys' school. This means that every element a in set A is a boy. This fundamental characteristic of set A is crucial for analyzing the given relations.
Step 2: Analyze Relation R
For the relation R, an ordered pair (a, b) belongs to R if a is the sister of b. Since set A contains only boys, no student a in A can be a sister to any other student b in A. Therefore, there are no pairs (a, b) that satisfy this condition.
Step 3: Conclude for Relation R
Since no pair (a, b) from set A can satisfy the condition that a is the sister of b, the relation R contains no elements. This means R is an empty relation.
Step 4: Analyze Relation R'
For the relation R', an ordered pair (a, b) belongs to R' if the absolute difference between their heights is less than 3 meters. The height of any human being is typically between 0.5 meters (for a small child) and 2.5 meters (for a very tall adult).
Step 5: Evaluate height difference
The maximum possible difference between the heights of any two students would be less than 3 meters. For example, if one student is 0.5 meters tall and another is 2.5 meters tall, their difference is 2 meters, which is less than 3 meters. It is practically impossible for the height difference between any two students to be 3 meters or more.
Step 6: Conclude for Relation R'
Since the difference between the heights of any two students in the school will always be less than 3 meters, every possible ordered pair (a, b) from set A will satisfy the condition for R'. Therefore, R' is the universal relation, which includes all possible pairs from A.