3. A collection of messages uses the following symbols with the given relative frequencies: s: 3 t: 10 e: 8 a:7 m: 1 (a) Construct the optimal Huffman code for these frequencies. (b) Encode the string test using this code. (c) Calculate the lower and upper bound of the average code length. Compare to the actual code length.

Create an algorithm to discover a number that is not in a set of n real numbers given a set of n real numbers. Demonstrate that your method is optimum by giving a lower limit on the number of steps needed to solve the issue that is based on information theoretic principles.

How many forms of proof are there, exactly? A simple proof may be used to show that every sum of two even integers is itself even.

METHODS OF PROOF PROVE THE FOLLOWING: 1. By contradiction, prove that "for all real numbers x, if x is irrational, then -x is irrational. *Please provide a clear proof. THANK YOU.