Can you please answer 16,20,30,31,41

Thankyou.

18. Prove that the set of noneegative rational sambers is denumerable Subtraction on the set of nataral numbers is defined by sets is countable Prove that there are an uncountable number of total fanctions from N to (0, 11 21. A total fanction f from N to N is said to be rmpeating if fn)- fn+ D for some n N. Otherwise, f is said to be sonrepeating. Prove that there are an uncountable This operation is often called proper subtraction Give a recursive definition of proper subtraction asing the operations s and pred number of repeating functions, Also prove that there are an uncountable number of 22. A total function f fronn N to N is momatone incrasing f n)+ D) for all n e let X be afinne set. Give a recursive definition of as the operator in the definition. 36 set of subsets of x. Use union N. Prove that there are an uncountable number of monotone increasing functions. 23. Prove that there are uncountably many total functions from N to N that have a fixed .37. Give a recursive definition of the set of finite subsets of N. Use union and the saccessor point Sce Example 1.4.3 for the definitiom of a fixed point 24. A total function f from N to N is mearly identity if f(n)- I., or n + I for every s as the operators in the definition. Prve tha 2 + 5 + 8+ +11,-I)snOn + 1)/2 for all->0. n. Prove that there are uncountably many nearly identity functions. 25. Prove that the sct of real numbers in the interval [O, 1) is uncountable. Hint: Use the 39 Prove thot 1+2+2++-I for all n 20. 41. Prove that 3 is a factor of n-+3 for all n 0 42. Let P-(A. B) be a set consisting proposition diagonalization argument on the decimal expansion of real numbers. Be sure that each Prove 1 + 2"