Prove by induction that, for any sets A_1, A_2, ..., A_n De Morgen's law generalises to
Describe a non-recursive algorithm that takes a list of distinct integers a_1, a_2, ...., a_n and finds the sum of the primes in the list. Write your answer in pseudo-code or any well-known procedural language like Python, Java, C++, ..... You do not need to write a function to determine whether a number is prime. Assume it is part of your language. E.g. For the list 2, 3, 4, 5, 6, 7, your program should return 17 (because 2 +...
Let a_0 + (a_1)(x) + (a_2)(x^2) + ... = 1/(1 - x - x^2). Prove that the coefficients of a_n are the Fibonacci Numbers.
Fill in the code
Procedure BinaryPeak(a_1, a_2, ..., a_n: 1. t:= 1 2. j:= n 3. while (t < j) 4. m [j/2] 5. if 6.:= m + 1 7. else 8. j:= m 9. return t
Prove by induction that if A and B are finite sets, A with n elements and B with m elements, then A x B has nm elements. Also, prove by induction the corresponding results for k sets.
Problem 30. Prove that N, Z, Q and R are infinite sets. (HINT: Prove by induction on n that is f: NN then (3k N(Vj Nn)k> f(j). Then conclude that f cannot possibly be onto N. A similar strategy works for Z, gq and R as well.)
11. Recall one of De Morgan's laws for families of sets: NACH A = UAEGĀ Equivalently, for all positive integers n: Ain A2 n... An = ALU AU... An Using a proof by induction, prove the latter of the above statements.
11. Recall one of De Morgan's laws for families of sets: NACH A = UAEGĀ Equivalently, for all positive integers n: Ain A2 n... An = ALU AU... An Using a proof by induction, prove the latter of the above statements.
Problem 2: Proof of Laws Consider sets ? and ?, and: Prove the associative law ?∩(?∩?)=(?∩?)∩? by membership table.
18. Use induction to prove the following: For any set A, if IA] = n for some finite number n E N, then IP(A) 2"
Use induction on n...
5. Use induction on n to prove that any tree on n2 2 vertices has at least two vertices of degree 1 (a vertex of degree 1 is called a leaf).
5. Use induction on n to prove that any tree on n2 2 vertices has at least two vertices of degree 1 (a vertex of degree 1 is called a leaf).