Let R={(x, y) ∈ℤ ×ℤ |x ≡y 5}. Recall that we saw in Section 4.6 that R is an equivalence relatio

Let R={(x, y) ∈ℤ ×ℤ |x ≡y 5}. Recall that we saw in Section...

Key Concepts

Modular Arithmetic

Modular arithmetic is a system of arithmetic for integers where numbers 'wrap around' upon reaching a certain value, called the modulus. It is a fundamental tool in number theory and plays a significant role in problems involving equivalence relations defined by congruence conditions. Properties of modular arithmetic are essential in verifying whether operations such as taking powers or exponentials preserve equivalence across representatives.

Well-defined Functions on Quotient Sets

When defining a function on a quotient set, it is crucial that the function is well-defined; that is, the function's output must depend only on the equivalence class and not on the specific representative chosen from the class. This requires ensuring that if two representatives are equivalent, then their images under the function are also equivalent, thereby establishing unambiguity in the function's definition.

Quotient Sets

A quotient set is formed by taking a set and partitioning it into equivalence classes under an equivalence relation. Instead of working with individual elements, one works with the classes as a whole. This abstraction simplifies many problems by reducing them to questions about these classes, and it is a central idea when considering functions defined on or between quotient sets.

Equivalence Relations

An equivalence relation on a set partitions the set into distinct, non-overlapping subsets known as equivalence classes, where every element is equivalent to the others in its class according to a given criterion. This concept is foundational in many areas of mathematics, such as number theory and abstract algebra, since it allows one to study objects up to an equivalence relation rather than individually.

Transcript

00:01 In this problem of relation and function, we have to show that relation r is equivalence relation and relation r is given in the set a, a is a set in which x belongs to z and such that x is a value between 0 and 12.

00:20 So this is the set a and relation r is given y.

00:25 Say relation r is the first, so relation r is given y.

00:31 R is a relation between ordered pair a and b such that modulars of a minus b is multiple of 4.

00:43 Is multiple of 4.

00:49 So this is the relation.

00:51 Now we have to show that this is a equivalence relation.

00:55 So here, a is the value between 0 to 12.

01:00 So we can write as a is a set in which the values are from 0 to 12, like 0 ,1 ,000.

01:07 2, 3 up to this is 12 and now we can write r so r would be say this should be multiple of 4 so first we are taking 4 and 0 the first pair would be 4 and 0 so we can write it 4 and 0 and also we can take 0 and 4 so this would be 0 and 4 and now 1 5 so this would be 1 5 and 5 1 and 5 1 and then 2662, 26 and 62.

01:49 Similarly, this would be up to 129912 here 129 and 912 and then this would be here 84448 8448 and in the end this would be 12 minus 12 or we can say 12 12.

02:16 So this is the r.

02:19 This relation is said to be reflexive if ordered pair like x and x should also be there.

02:26 So which belongs to the given r.

02:29 So when we write x is equal to 0 2, say we are seeing 4, 5, 6, 7, 8, 9, 10.

02:36 So there would be always such pair such as this is like 7 minus 7 would be equals to 0.

02:44 So this is a multiple of 4, 8 minus 8, modules of 8 minus 8 is 0.

02:48 So we can say that relation r is reflexive.

02:56 And now we have to check for symmetric relation.

02:59 So suppose the ordered pair, a, b is in r, then ordered pair b and a should also be there.

03:09 So as you can know that here, 4 -0 is there, that means 0 -4 is also there.

03:13 1 and 5 is there, 5 and 1 is there, 2 and 6 are there, and 6 and 2 are there...

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