Empty and Universal Relations
30 Questions
0 Views

Choose a study mode

Play Quiz
Study Flashcards
Spaced Repetition
Chat to lesson

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

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.

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

Explain what it means for a relation to be symmetric.

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

Define a transitive relation using ordered pairs.

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

What is the definition of an invertible function?

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

How can you prove that a function is invertible?

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

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

<p>f(x) = 4x + 3</p> Signup and view all the answers

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

<p>g(y) = Rationalised 2023-24 (y - 3)</p> 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?

<p>Yes</p> Signup and view all the answers

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

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

Which integers are related to zero in the relation R?

<p>Even integers</p> Signup and view all the answers

Which integers are related to one in the relation R?

<p>Odd integers</p> Signup and view all the answers

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

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

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

<p>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.</p> Signup and view all the answers

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

<p>E</p> 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.

<p>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.</p> 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?

<p>Yes</p> 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}}?

<p>Yes</p> Signup and view all the answers

What properties make a relation an equivalence relation?

<p>Reflexive, symmetric, and transitive</p> Signup and view all the answers

How can you show that a relation is reflexive?

<p>(x, y) R (x, y) for all (x, y) in the set</p> Signup and view all the answers

Explain how to show symmetry in a relation.

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

What does it mean for a relation to be transitive?

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

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

<p>Leibnitz</p> Signup and view all the answers

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

<p>John Bernoulli</p> Signup and view all the answers

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

<p>Leonhard Euler</p> Signup and view all the answers

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

<p>1793</p> Signup and view all the answers

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

<p>Lejeunne Dirichlet</p> Signup and view all the answers

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

<p>Georg Cantor</p> Signup and view all the answers

More Like This

Use Quizgecko on...
Browser
Open
Browser
Browser