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Questions and Answers
Which of the following sets are related under the relation R defined by |a - b| is even?
The relation R defined by R = {(L1, L2) : L1 is parallel to L2} is reflexive.
True
What is an example of a relation that is symmetric but neither reflexive nor transitive?
R = {(1, 2), (2, 1)}
The relation R defined on triangles, where T1 is similar to T2, can be deemed __________.
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Match the type of relation with its properties:
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Which of the following is an example of a relation from the set of students in Class XII to Class XI?
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In mathematics, a relation is always a connection based on a recognisable link.
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What is the range of a relation?
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A relation that contains no elements is called the ______.
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Match the types of relations with their descriptions:
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Study Notes
Relation Definitions
- A relation between sets A and B is a subset of A × B.
- Empty relation: No element of A is related to any element of A, represented as φ.
- Universal relation: Each element of A is related to every element of A, represented as A × A.
- Reflexive relation: Every element 'a' in set A is related to itself, (a, a) ∈ R.
- Symmetric relation: If element 'a1' is related to 'a2', then 'a2' is related to 'a1', (a1, a2) ∈ R implies (a2, a1) ∈ R.
- Transitive relation: If 'a1' is related to 'a2' and 'a2' is related to 'a3', then 'a1' is related to 'a3', (a1, a2) ∈ R and (a2, a3) ∈ R implies (a1, a3) ∈ R.
- Equivalence relation: A relation that is reflexive, symmetric, and transitive.
Function Definitions
- A function is a special type of relation where each element in the domain maps to exactly one element in the codomain.
- One-one (injective) function: Distinct elements in the domain map to distinct elements in the codomain, f(x1) = f(x2) implies x1 = x2.
- Many-one function: At least two elements in the domain map to the same element in the codomain.
- Onto (surjective) function: Every element in the codomain is the image of at least one element in the domain, for every y in Y, there exists an x in X such that f(x) = y.
- Bijective function: A function is both one-one and onto.
Examples
- Example 1: In a boys' school, the relation "a is sister of b" is the empty relation, while the relation "the difference between heights of a and b is less than 3 meters" is the universal relation.
- Example 2: The relation "T1 is congruent to T2" in the set of all triangles is an equivalence relation.
- Example 7: The relation "f(x) = roll number of student x" in the set of all students of Class X is a one-one and onto function.
Function Historical Development
- The concept of function has evolved over time, starting with Descartes who used the word "function" to describe positive powers of a variable.
- Gregory expanded this to include quantities derived through algebraic operations, while Leibnitz defined functions as quantities depending on variables.
- John Bernoulli introduced the notation φx for functions, later standardized by Euler.
- Lagrange discussed analytic functions and used the notation we recognize today, like f(x) and F(x).
- Dirichlet provided a more rigorous definition of a function which was further refined with the development of set theory.
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Description
Test your understanding of relations and functions in set theory. This quiz covers key concepts such as empty relations, equivalence relations, and different types of functions including injective functions. Ideal for students learning set theory or preparing for exams.
