Which of the following are logically equivalent to the statement "Some fuzzy rabbits do not have floppy ears." if U is the set of all rabbits, p(x) is the statement "x is fuzzy" , q(x) is the statement "“x has floppy ears", F is the set of all fuzzy rabbits, and E is the set of all rabbits with floppy ears? (Choose all that apply): I. FNE° + Ø. II. (Vx E U)(¬(p(x) → q(x))). III. Some rabbits with floppy ears are not fuzzy. IV. F°n E + Ø. V. Not all fuzzy rabbits have floppy ears. VI. (3x E U)(p(x) ^ ¬q(x)). VII. F°U E + U.

In N × N clause β for (a,b), (c,d) ∈ N × N defined as:(a,b)β(c,d) = ad = bcExamine the properties of the relation β.

1. Use set identities to show that (A U B)n (BU A) = A where A and B are sets. Let A = {1,2, 3} and B = {{0}, {1}}. Find A × B. 2.

For all relations R and S, Ro S = So R. O True O False

Let G = {-23, –7, -3, –1,1,3, 5, 7, 14, 27, 42}. Decide if the following statements are true/false. Give a support to your answer. (a) Vr E G : if I 7. (d) 3r, y E G : (r² - y² = 0) ^ (x > 10) ^ (y > 10).

In Real analysis

Q11: If X and Y are denumerable sets, then X - Y is countable.

Q11: The symmetric difference of A and B is denoted by A + B which is defined by A + B = (A - B) U (BA): For all sets A; B and C, prove the following: 1- A + C = A if and only if C = Ø 2- A + C = Ø if and A = C

3. Translate the following symbolic sentences into English. Make your answer as concise as possible. (a) \n € N, 2n+1 x. (c) a € R, Vx € R,x+|x|a. Which, if any, of these statements are true?