Master Mathematician
Its all about Mathematics.
Tuesday, 28 July 2020
Important result related congruence relation and an example.
The proof is in video link given below.https://youtu.be/fugV55xPq4g
Theorem 4.3. If ca= cb (mod n), then a= b (mod n/d), where d = gcd(c, n).
Theorem 4.3. If ca= cb (mod n), then a= b (mod n/d), where d = gcd(c, n). Proof. By hypothesis, we can write c(a - b) = ca - cb = kn
for some integer k. Knowing that gcd(c, n) = d, there exist relatively prime integers r ands satisfying c = dr, n = ds. When these values are substituted in the displayed equation and the common factor d canceled, the net result is
r(a - b) =ks
Hence, s I r(a - b) and gcd(r, s) = 1. Euclid's lemma yields s I a - b, which may be recast as a = b (mods); in other words, a = b (mod n/d).
For video lec link is below.
https://youtu.be/dopXp7SyKNo
for some integer k. Knowing that gcd(c, n) = d, there exist relatively prime integers r ands satisfying c = dr, n = ds. When these values are substituted in the displayed equation and the common factor d canceled, the net result is
r(a - b) =ks
Hence, s I r(a - b) and gcd(r, s) = 1. Euclid's lemma yields s I a - b, which may be recast as a = b (mods); in other words, a = b (mod n/d).
For video lec link is below.
https://youtu.be/dopXp7SyKNo
a is congruent to b(modn) if and only if a and b give same positive remainder upon division by n
The prove is very simple and straight forward you can get in video link below.
Video link.
https://youtu.be/RW0QD8van60https://youtu.be/RW0QD8van60
Video link.
https://youtu.be/RW0QD8van60https://youtu.be/RW0QD8van60
Congruence Relation
Definition 4.1. Let n be a fixed positive integer. Two integers a congruent modulo n, symbolized by
a= b (mod n) if n divides the difference a b; that is, provided that a b = kn for some integer k.
consider n = 7. It is routine to check that 3 = 24 (mod 7)
- 31 = 11 (mod 7) , -15 = -64 (mod 7) because 3 - 24 = (-3)7 -31- 11 = (-6)7, and -15 -(-64) = 7 · 7. When
n A' (a - b), we say that a is incongruent to b modulo n, and in this case we write a ¢ b (mod n). For a simple example: 25 ¢ 12 (mod 7), because 7 fails to divide 25 -12 = 13. It is to be noted that any two integers are congruent modulo .
https://youtu.be/BzZ6fJ4KWAU
Wednesday, 22 July 2020
Derivative of inverse function derivation
Here i derived derivative of inverse function of sinx and cosx.
Hope you understand and got the idea i actuallu want to share with you people.
Any suggestions .
Question if any you may ask friendly.
Set theory results.
This is an important theorem related to set theory. If any question or suggestions kindly comment.i will answer you.
If you have any question relating set theory u can ask freely
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Important result related congruence relation and an example.
The proof is in video link given below. https://youtu.be/fugV55xPq4g