Relations and Functions

Introduction to Relations and Functions

• Relations and functions will be covered in this lesson.
• Some students admit they haven't studied the topic well throughout the year.
• The lesson is designed to be understandable even for those who haven't studied the topic in depth.
• A set, denoted as A, is introduced with elements 1, 2, and 3.
• Set B is mentioned with elements 4 and 5.
• A relation, denoted as R, is described using x and y, where x is from set A and y is from set B.
• The relation R is defined such that x + y > 3.
• Ordered pairs are formed based on the provided relation where x is taken from set A, and y is taken from set B such that x + y > 3
• The subsets that meet the relationship criteria are called order pairs.

Types of Relations

• The conversation mentions a plan to cover the entire chapter in a simple manner.
• It's noted some students may only be familiar with 3 types of relations but there are 5.
• The types are listed as empty, full, reflexive, symmetric, and asymmetric.

Definition of Empty Relation

• "Empty relation" is the same as a "null set" or "empty set."
• Using the set A containing 1, 2, 3, it's explained that in A cross A, elements are chosen from the set itself.

Definition of Universal Relation

• Universal Relation: using all elements from the set with no extra conditions
• Set A is expressed again for clarity: A = {1, 2, 3}
• A relation is formed: x + y > 0
• This relation includes any possible pair because it covers all the elements in order to satisfy the initial condition.
• A "universal relation" would completely use up all the components/ elements to generate a set.

Basics of Reflexive Relation

• Reflexive Relation: in this kind of relation, there needs to be a doublet
• If we have set A consisting of elements 1, 2, 3, 4 then all the double relationships must be there when creating a set. (1,1), (2,2), (3,3), (4,4)

Additional Notes on Reflexive Relation

• It should have doubles of all the elements in the set
• The presence of extra elements beyond the doubles doesn't affect whether the relation is reflexive.

Basics of Symmetric Relation

• To explain the symmetric relation, the condition is that if (a1, a2) belongs to R, (a2, a1) also must belong to R.
• If a1 and a2 show some relationship, then in order, once a2 comes first, then a1 the relationship needs to exist.

Transitive Relation

• Transitive Relation: If (a1, a2) belongs to R and (a2, a3) belongs to R, then (a1, a3) must belong to R.
• An analogy with sets of friends is used where Amit is friends with Sumit, and Sumit is friends with Akshay. Therefore, Amit is also a friend with Akshay

Equivalence Relation

• Equivalence Relation: a relation is considered a combination of reflexive, symmetric, and transitive relations.
• If a relation satisfies all three properties, it is an "equivalence relation."

Question 1

• The relation set, denoted as R, in set A is defined.
• Set A is defined as: A = {1, 2, 3, 4, 5, 6}.
• The explicit relationship is: x and y such that y is divisional by x.
• Must explain:
  • Reflexive: Yes or no
  • Symmetric: Yes or no
  • Transitive: Yes or no

Solution to Question 1: Testing for Reflexive

• Reflexive: a relation is reflexive if a relates to itself
• Any number will always be divisible by each other
• "A divisible by a"
• Given the above logic, the given relation is reflexive.

Solution to Question 1: Testing for Symmetric

• The definition of symmetrical is explained using "if a1, a2 belongs to relation R then a2, a1 must also belong to relation R".
• Full example: given y has to be dividable by x, 2 divisions by 1 is true. However, 1 can't completely undergo division by 2 in order for the relationship to be symmetric.
• The relationship is not symmetric.

Solution to Question 1: Testing for Transitive

• Definition of transitivity: if a1 and a2 belongs to R and a2 a3 belongs to R, then a1 to a3 will also be true.
• Example: y is divisible by x AND z is divisible by y, therefore we can say that x, z also belong to R since divisible can happen.
• Overall Analysis: only reflexive, only transitive relation

Question 2

• Relation R to use integer as its set
• Integers: negative infinity to positive infinity.
• Condition: x - y needs to also produce integer (-inf to inf).
• Requirements: Reflexive, Symmetric, and Transitive.

Solution to Question 2: Testing for Reflexive

• "a, a belongs to relation R because x - x = 0 and all values end as numbers, therefore reflexive"

Solution to Question 2: Testing for Symmetric

• Symmetric: (a1, a2) belongs to R, then (a2, a1) will belongs to R.
• If x - y = k, then y - x = -k where = k = integer.
• Relation is symmetric.
• An example used to better demonstrate this, if we use elements within the sets of integers, whether subtracted normally or conversely, it will render an integer.

Solution to Question 2: Testing for Transitive

• Requirements for Transitive = (a1, a2) belongs to R AND (a2, a3) belongs to R THEN (a1, a3) belongs to R
• if x - y = k and y - z = r, x - z = k + r. and since these elements k and r are integers the results, x - z remains an integer.
• an integer divided by another integer still yields an integer.
• Example of adding in for a visual demonstration

Question 3

• A is equal to B squared and is either less than or equal.
• Requirements:
  • What is Reflexive
  • What is symmetric?
  • What is Transitive

Solution to Question 3: Testing for Reflexivity

• Not reflexive.
• a,a as relation, we can apply 1 to 2, but not when we square it
• The explanation for this is that after 1/2 squared, the equation becomes 1/2 is either less than or equal to 1/4 which doesn't work out.

Solution to Question 3: Testing for Symmetry

• Explain test (a1, a2) belongs to R, (a2, a1) belongs to R relationship
• Example: using the numbers 1 and 4. The first relationship belongs, but the second doesn't because if 4 was in the A position, and one was in the B position, that could never belong.

Solution to Question 3: Testing for Transitivity

• Testing definition:
  • if a1, a2 and a2, a3 belong to set. then a1, a3 will belong.
  • Example: using number 8, 4 belong, and using 4, 2 belonging. This could indicate a trans relationship.
    • HOWEVER, after combining, we get invalid because "8, 2 doesn't belong.
    • the test for a1, a3 isn't true, therefore can't be true.

Question 4

• a to b relationship where a - b = even, a and b need to satisfy such relation
• Is also a Equivalence relationship?

Solution to Question 4: Testing Reflexively

• Because a - a = 0. Even relationship is true given previous conditions.

Solution to Question 4: Testing Symmetrically

• if a1, a2 belongs, then a2 and a1 must belong
• Take 3 and 1 for example, use absolute value, take another value or two like 2 or 4 to prove other examples.

Solution to Question 4: Testing Transitivity

• Defintion: when we see a1 a2 and the a2 a3 both show true relationships. then finally a1, a3 are going to be true so.
• Test: a - b = k AND b - c = r THEN a - c = something to get the rest.

Additional Notes on Question 4

• Also needs to show that elements with 1 3 or 5 are linked and that 2 and 4 are linked
• To show this, a - b is linked within 1 3 and 5 relationships and another set b 2 and four have also.

Question 5

• In a plane such that P and Q are related.
• the distance of P from the Origin is the same as the distance from Q.

Solution to Question 5: Testing Reflexivity

• Testing reflexivity has the same logic where the distance needs to be from origin.
• P, p belongs with since from its origin it belongs because measuring twice or more doesn't make.

Solution to Question 5: Testing Symmetry

• P, Q belongs since p distance, when we do the Q first and then the q, we will show same relationship from it distance to origin
• The condition of how much is a reflection.

Solution to Question 5: Testing Transitivity

• A new one Z has with distance. Then this transitive where Z gets new relationship that continues.

Question 6

• L be all sets in line in plain
• "parallel (l1,l2) then must parallel. How equalance (show if each of three tests is equal)
• Show l1 l1 has same line is parallel. L1 always parallel itself
• l1 parallel to l2 and does a reversal in the order. If it exists, then it could work and is in turn is also called the symmetrical property exists.

Practice Question Insight/Tips

• If something is a true condition then you should use definition
• If not prove provide examples.

Functions

• Function = machine
• Input and output.
• F(x) equal with x squared.
• X can be one.
• One sqaured one
• Number two
  • Then # sqaure then output it's four
  • Number 3 equal nine 4 = 16 for square.

Concepts To Consider/Understand Regarding Functions

• In mathematics, focus is on types of functions: sets, range
• Set side and another sets side with codomains
• Coding is something related to book

Many-to-One vs One-to-One (Example)

• Person named sonu
• With sisters Neala
• All have diff sisters named
• All have one brother

What is On2 (How to Do)

• In all codmain needs to be linked = then call one2. Then must have a relationship.
• In relation then that is called all the items that book end and then range are all called booked then that's call one2. Then must have a relations as well.

Mathematical Explanation

• With each definition that are true call equal one to one. And each input equal each other. Then call one to one.

Question 7

• N to N. Meaning from neutral to nutrual, with natural numbers to one and two
• Function, you are not to make a number or an expression to be what it can't be. Do an equation and not assume.
• There exists only
• What you will be reading. N

Questions

• Show functions r to are with "x minus 2 ,with x minus 3 with it is 2 and 2" then "1 point" to show how it isn't

Solution

• 1/1 and all those numbers one one to one so that's how know it goes
• Take definition, with fx one equals fx two two then x sub one and all the other one you see will become numbers and equations etc.
  • You'll want to cross milt and solve the equation with plus 6 and all equal.

What is Coding of Comdomain with X and Set One With Other Set

• Coding will happen and each set happens for one is equal at that end with everything there, it's called functions

Steps To Understanding How It Is Called One 2.

• with x will have what it means for that, not to be that it can't be.

More Complex Practice Questions Regarding 2 and

• Great integer function - the example shows greatest with point and also.

Question

• With greatest function with r to r, then it is one with -x it needs to be x squared (to solve if each is both or etc).