In this exercise we will check if "f" is a linear transformation. It maps from and to a polynomial vector space and has a residue class as its field. ⏰ Timeline 00:00 Exercise 00:38 Residue class field 01:18 Showing linearity 03:18 Comparing results 04:04 Conclusion ? All Discrete Mathematics Exercises https://www.youtube.com/playlist?list=PLY9Po-aXYcD6LdOzLeBhcHIShPwCQNeSD ? All Linear Algebra Exercises https://www.youtube.com/playlist?list=PLY9Po-aXYcD5BnL_9CcYy421JLvwn9XHH ? Music Reverie by Nomyn https://soundcloud.com/nomyn Creative Commons — Attribution 3.0 Unported — CC BY 3.0 Free Download / Stream: http://bit.ly/2RM3qu4 Music promoted by Audio Library https://youtu.be/LRNX-lgE8mo ... https://www.youtube.com/watch?v=VfchRkl3Z7k

In this exercise we need to proof by induction over that the the power set of a set with n elements has exactly 2^n elements or we could also say that it has the cardinality of 2^n. ⏰ Timeline 00:00 Exercise 00:10 Base case 00:50 Induction hypothesis 01:03 Induction step 03:46 Conclusion ? All Discrete Mathematics Exercises https://www.youtube.com/playlist?list=PLY9Po-aXYcD6LdOzLeBhcHIShPwCQNeSD ? All Linear Algebra Exercises https://www.youtube.com/playlist?list=PLY9Po-aXYcD5BnL_9CcYy421JLvwn9XHH ? Music Reverie by Nomyn https://soundcloud.com/nomyn Creative Commons — Attribution 3.0 Unported — CC BY 3.0 Free Download / Stream: http://bit.ly/2RM3qu4 Music promoted by Audio Library https://youtu.be/LRNX-lgE8mo ... https://www.youtube.com/watch?v=FPwByuRMJwU

In the 4th exercise of the "Discrete math" series we will find out for which n the Josephus number is n/2. We will find the conditions and afterwards proof a special requirement by induction. ⏰ Timeline 00:00 Exercise 00:14 Josephus problem 00:58 Exercise goal 01:30 Finding requirements 03:06 Why we are not done yet 03:23 When is r a whole number? 04:46 Proof by induction 08:19 Conclusion 08:37 Example ? Useful links https://youtu.be/uCsD3ZGzMgE ? Formula J(n) = n / 2 ?️ Thumbnail icon credits Collaboration icon made by mynamepong from www.flaticon.com Swords icon made by Freepik from www.flaticon.com ... https://www.youtube.com/watch?v=fz_P7a8XaYc

In this exercise we will cover all 9 field axioms for a tuple of real numbers with a special addition and multiplication. ⏰ Timeline 00:00 Exercise 00:15 Commutativity of addition 00:45 Commutativity of multiplication 01:08 Associativity of addition 01:48 Associativity of multiplication 03:10 Additive identity 03:38 Multiplicative identity 04:27 Additive inverse 05:05 Multiplicative inverse 05:56 Distributivity 07:05 Conclusion ? Ring definition R x R := {(a,b) | a,b in real numbers } (a,b) + (c,d) := (a+c, b+d) (a,b) * (c,d) := (a*c - b*d, a*d + b*c) ? All Discrete Mathematics Exercises https://www.youtube.com/playlist?list=PLY9Po-aXYcD6LdOzLeBhcHIShPwCQNeSD ? All Linear Algebra Exercises https://www.youtube.com/playlist?list=PLY9Po-aXYcD5BnL_9CcYy421JLvwn9XHH ?️ Thumbnail icon credits Packing List icon made by Freepik from www.flaticon.com Field icon made by Freepik from www.flaticon.com ? Music Hold On by Joakim Karud https://soundcloud.com/joakimkarud Creative Commons — Attribution-ShareAlike 3.0 Unported— CC BY-SA 3.0 Free Download / Stream: https://bit.ly/_hold-on Music promoted by Audio Library https://youtu.be/Ks1rRmClD9Y ... https://www.youtube.com/watch?v=cTUtbR4avFg