(p ⇒ q) ∨ (p ∧ ~ q)
Mel: "I am a knight and Zeke is a knave."Is Mel a knight or a knave? Is Zeke a knight or a knave? Show your reasoning.
Zeke: "Mel is a knight."
As in the Three Cannibals and Three Missionaries Problem, three women and their three boyfriends must cross a river using a boat which holds only two people. Here, the constraint is that no boyfriend is left with one or both of the other women without his own girlfriend also present.Is a solution possible? Formulate this problem as an undirected graph and either present a solution or argue that none exist.
It is not true that if the aliens are green then they either have three heads or else they cannot fly, unless it is also true that they are green if and only if they can fly and that they do not have three heads.Assume that the aliens are all alike. Let f be the proposition that "the aliens can fly", let g be the proposition that "the aliens are green" and let h be the proposition that "the aliens have three heads".
Use logic notation and the variables f, g and h to rephrase the space traveler's report.
Can the aliens fly? Are they green? Do they have 3 heads? Show your work.
If n^{5} + 7 is an odd integer, then n is an even integer.
x ⋅ y ≡ 1 (mod 17).In other words, x ⋅ y % 17 = 1. Then, x and y are called inverses modulo 17.
x ≡ 4 (mod 5)Note that 5 ⋅ 9 ⋅ 14 = 630 and recall that the notation
x ≡ 2 (mod 9)
x ≡ 7 (mod 14)
a ≡ b (mod n)means that a % n = b % n, where % is the remainder operator.
a^{ p  1} ≡ 1 (mod p).Take advantage of Fermat's Little Theorem to compute the value 9^{5282} % 17 by noticing that 9^{16} ≡ 1 (mod 17). Show your work.
A ∩ (B ⊕ C) = (A ∩ B) ⊕ (A ∩ C)A wellwritten proof must include complete English sentences describing major steps of the proof.
(f _{°} g) (x) = f (g(x)).Show that if f is onetoone and g is onetoone then f _{°} g is onetoone.

+ 

+ 

+ ⋅⋅⋅ + 

= 

. 
(1 − 2 ^{−2}) ⋅ (1 − 3 ^{−2}) ⋅ (1 − 4 ^{−2}) ⋅⋅⋅ (1 − n ^{−2})  = 

. 
1 + nx ≤ (1 + x)^{n}.Note: the inequality actually holds for x ≥ −1.
16, 22, 12, 8, 15, 11, 10, 7, 14, 5, 9, 24and with n equal 12. How is A rearranged when partition returns? What is the meaning of the value returned by partition? (Hint: the partition() function might be used in Quicksort.)
Note: a relation R ⊆ A × B is antisymmetric if for all (a, b) ∈ A × B, (a, b) ∈ R and (b, a) ∈ R implies that a = b.
Turns out this problem cannot be done without knowing the characters in the banned words. For example, if OOOO is banned, then AOOOOBC and ABOOOOC are both banned. These are easy to account for, but we must also make sure that AOOOOOB (with 5 O's) is counted only once. To do this we need to know which of the 61 words are "self overlapping".