Problem 4 (2pts) Let A, B be formulas B) (-AV B) Give a formal proof for...
Problem 3 (3pts) ment "x is a class", let S(x) be the statement " x is a student" and let I(x,y) be the statement "x is in y". Translate the followings into formulas. Let P(x) be the statement "x knows functional programming", let C(x) be the state- Every class has a student who knows functional programming. Every student in every class knows functional programming. There is at least one class with no students who know functional programming. Problem 4 (2pts)...
Give a formal proof for the following tautology by using the CP rule. (B →C) ^ A ^ B →A ^ C
Number 3 +4n + 8 is O(n). 3. Give a formal proof that f(n) 5m3 +3n2 4. Give a formal proof that f(n)-7*2n+ 9m3 is O(2n). 5. Give a formal proof that log (n + 1) is O(log n).
Complete the proof. [}proof{]; 1.A>B, premise; 2. Av~B, premise]:A=B Answer: Dproof(;
PLEASE PUT SOLUTION IN THE FORM OF A FORMAL PROOF. Let (X, d) be a metric space and give R the usual Euclidean metric. Assume that f:XR is continuous. (a) S the set {x E X | f(x) a} is open for all a R. (b) Show that the set (xEXIf(x) a is closed for all a E R. how thait
Give a formal proof of a valid argument. if not valid then give a counter example. if Oscar attends class, then so does Miriam, and if Miriam attends class, so does George. Oscar attends class unless George attends. Therefore, Miriam does not attend class (O, M, G).
For each of the following, use Fitch to give a formal proof of the argument. These look simple but some of them are a bit tricky. Don't forget to first figure out an informal proof. Taut Con and FO Con cannot be used. ∀x (Small(x) → Cube(x)) ∃x ¬Cube(x) → ∃x Small(x) - ∃x Cube(x)
Formal proof and state which proof style you use Let a function where f:Z5 → Z5 defined by f(x) = x3 (mod5). a. Is f an injection? Prove or provide a counter example. b. Is fa surjection? Prove or provide a counter example. c. Find the inverse relation of f. Verify that it is the inverse, as we have done in class. d. Is the inverse of f a function? Explain why it is or is not a function.
Problem 1. (4 pts) Combinatorics and the Principle of Inclusion Exclusion (a) (2pts) Roll a fair die 10 times. Call a number in 1, 2, 3, 4, 5, 6 a loner if it is rolled exactly once on the 10 rolls. (For example, if the rolls are 1 2 6 4 4 4 6 3 4 1, then 2 and 3 are the only loners) Compute the probability that at least one of numbers 1, 2, 3 is a loner....
25. Give a formal proof relative to why it is the case that, under the hypothesis XY-5, XZ 3, and m(Y) 45°, we would no longer be able to construct a triangle AXYZ no matter how hard we try.