Empty and Universal Relations

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Explain why the relation R' is considered a universal relation.

The relation R' is considered a universal relation because the difference between heights of any two students of the school is always less than 3 meters.

Define an empty relation using set notation.

An empty relation is a relation that has no elements. In set notation, it is represented as R = φ.

How can a relation be represented in set builder notation?

A relation can be represented in set builder notation as R = {(a, b) : some condition relating a and b}.

Define a reflexive relation in terms of ordered pairs.

A relation R in a set A is reflexive if (a, a) ∈ R for every element a ∈ A. Signup and view all the answers

Explain what it means for a relation to be symmetric.

A relation R is symmetric if for any (a1, a2) ∈ R, then (a2, a1) ∈ R as well. Signup and view all the answers

Define a transitive relation using ordered pairs.

A relation R is transitive if for any (a1, a2) and (a2, a3) in R, then (a1, a3) must also be in R. Signup and view all the answers

What is the definition of an invertible function?

A function f : X → Y is invertible if there exists a function g : Y → X such that gof = IX and fog = IY. Signup and view all the answers

How can you prove that a function is invertible?

To prove a function f is invertible, it must be shown that f is both one-one and onto. Signup and view all the answers

In Example 17, what is the function f(x) defined as?

f(x) = 4x + 3 Signup and view all the answers

What is the inverse of the function f(x) = 4x + 3 in Example 17?

g(y) = Rationalised 2023-24 (y - 3) Signup and view all the answers

Is the relation R in the set Z of integers given by R = {(a, b) : 2 divides a – b} an equivalence relation?

Yes Signup and view all the answers

How is the invertibility of a function shown in Example 17?

The function f is invertible as gof = IX and fog = IY. Signup and view all the answers

Which integers are related to zero in the relation R?

Even integers Signup and view all the answers

Which integers are related to one in the relation R?

Odd integers Signup and view all the answers

In Example 18, why is R1 ∩ R2 reflexive if R1 and R2 are equivalence relations?

R1 ∩ R2 is reflexive because (a, a) belongs to both R1 and R2 for every element a in A. Signup and view all the answers

What are the conditions satisfied by the sets E (even integers) and O (odd integers) in relation R?

i) All elements of E are related to each other and all elements of O are related to each other. ii) No element of E is related to any element of O and vice-versa. iii) E and O are disjoint and Z = E ∪ O. Signup and view all the answers

What is the equivalence class containing zero denoted as in the relation R?

E Signup and view all the answers

Explain why the set E of even integers and the set O of odd integers are disjoint in relation R.

All even integers are related to zero, while all odd integers are related to one. Since zero and one are distinct, the sets E and O are disjoint. Signup and view all the answers

Is the relation R defined by (x, y) R (u, v) if and only if xv = yu an equivalence relation?

Yes Signup and view all the answers

For the relation R1 in X, where R1 = {(x, y) : x – y is divisible by 3}, is it equal to R2, where R2 = {(x, y): {x, y} ⊂ {1, 4, 7} or {x, y} ⊂ {2, 5, 8} or {x, y} ⊂ {3, 6, 9}}?

Yes Signup and view all the answers

What properties make a relation an equivalence relation?

Reflexive, symmetric, and transitive Signup and view all the answers

How can you show that a relation is reflexive?

(x, y) R (x, y) for all (x, y) in the set Signup and view all the answers

Explain how to show symmetry in a relation.

If (a, b) is in the relation, then (b, a) must also be in the relation Signup and view all the answers

What does it mean for a relation to be transitive?

If (a, b) and (b, c) are in the relation, then (a, c) must also be in the relation Signup and view all the answers

Who was the first to use the phrase 'function of x'?

Leibnitz Signup and view all the answers

Which mathematician used the notation φx for the first time in 1718?

John Bernoulli Signup and view all the answers

Who is credited with the general adoption of symbols like f, F, φ, ψ to represent functions?

Leonhard Euler Signup and view all the answers

In which year did Joseph Louis Lagrange publish his manuscripts 'Theorie des functions analytiques'?

1793 Signup and view all the answers

Who gave the definition of function that was used until the set theoretic definition was developed?

Lejeunne Dirichlet Signup and view all the answers

Who developed the set theoretic definition of function after the development of set theory?

Georg Cantor Signup and view all the answers

Empty and Universal Relations

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Questions and Answers

Explain why the relation R' is considered a universal relation.

Define an empty relation using set notation.

How can a relation be represented in set builder notation?

Define a reflexive relation in terms of ordered pairs.

Explain what it means for a relation to be symmetric.

Define a transitive relation using ordered pairs.

What is the definition of an invertible function?

How can you prove that a function is invertible?

In Example 17, what is the function f(x) defined as?

What is the inverse of the function f(x) = 4x + 3 in Example 17?

Is the relation R in the set Z of integers given by R = {(a, b) : 2 divides a – b} an equivalence relation?

How is the invertibility of a function shown in Example 17?

Which integers are related to zero in the relation R?

Which integers are related to one in the relation R?

In Example 18, why is R1 ∩ R2 reflexive if R1 and R2 are equivalence relations?

What are the conditions satisfied by the sets E (even integers) and O (odd integers) in relation R?

What is the equivalence class containing zero denoted as in the relation R?

Explain why the set E of even integers and the set O of odd integers are disjoint in relation R.

Is the relation R defined by (x, y) R (u, v) if and only if xv = yu an equivalence relation?

For the relation R1 in X, where R1 = {(x, y) : x – y is divisible by 3}, is it equal to R2, where R2 = {(x, y): {x, y} ⊂ {1, 4, 7} or {x, y} ⊂ {2, 5, 8} or {x, y} ⊂ {3, 6, 9}}?

What properties make a relation an equivalence relation?

How can you show that a relation is reflexive?

Explain how to show symmetry in a relation.

What does it mean for a relation to be transitive?

Who was the first to use the phrase 'function of x'?

Which mathematician used the notation φx for the first time in 1718?

Who is credited with the general adoption of symbols like f, F, φ, ψ to represent functions?

In which year did Joseph Louis Lagrange publish his manuscripts 'Theorie des functions analytiques'?

Who gave the definition of function that was used until the set theoretic definition was developed?

Who developed the set theoretic definition of function after the development of set theory?

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