Student ID: dent Name: Let c(x) mean x is a criminal, let r(x) mean x is rich, a
ID: 3340668 • Letter: S
Question
Student ID: dent Name: Let c(x) mean x is a criminal, let r(x) mean x is rich, and let s(x) mean x is sane. Find a wff to describe each sentence over the domain of people Every criminal is rich. Solution: a. b. Some criminals are neither rich nor sane Solution c Not all criminals are sane. Solution (x + y) mod 4. Answer each of Let ° be defined over the set {0.1.2.3) by xy the following questions. ) a. Does have a zero? Solution b. Does have an identity? If so what is the identity? Solution What elements, if any, have inverses? Solution c. d. Is o commutative? Solution B+ associative? Explain why? Solution eExplanation / Answer
(According to Chegg policy, only one question with multiple subquestions will be answered. Please post the remaining in another question)
x . y = (x + y) mod 4
a. We have x .0 = (x + 0) mod 4
= x mod 4
Thus there is a zero and that is {0}.
b. We have x .0 = (x + 0) mod 4
= x mod 4
Thus there is an identity and that is 0.
c. We have (0 + 4) mod 4 = 4 mod 4 = 0
(1 + 3) mod 4 = 4 mod 4 = 0
(2 + 2) mod 4 = 4 mod 4 = 0
(3 + 1) mod 4 = 4 mod 4 = 0
(4 + 0) mod 4 = 4 mod 4 = 0
=> All elements have inverses.
The inverse of 1 is 3 and vice-versa. The inverse of 0 is 4 and vice-versa. The inverse of 2 is 2.
d. Since x . y = (x + y) mod 4 = (y + x) mod 4 = y . x, . is commutative.
e. x . (y . z) = (x + (y + z) mod 4) mod 4 = ((x + y) mod 4 + z) mod 4.
Therefore, . is associative.