Provide complete definitions for the following:
Provide complete definitions for the following: 7 marks] Choosing a universe and predicates. (a) Consider the...
4. [7 marks] Choosing a universe and predicates. (a) Consider the following statement: Vx EN, P(r, 165)P(x,1) Provide one definition of a binary predicate P over Nx N that makes the above statement True, and another definition of P that makes the statement False. Briefly justify your answers, but no formal proofs are necessary b) Consider the following statement: Provide one definition of a non-empty set U, and predicates P, Q, and R over U, that makes the above statement...
Consider the following statement Provide one definition of a non-empty set U, and predicates P, Q, and R over U, that makes the above statement true, and another definition of a non-empty set U, and predicates P, Q, and R that makes the statement false. Briefly justify your answers, but no formal proofs are necessary. Consider the following statement Provide one definition of a non-empty set U, and predicates P, Q, and R over U, that makes the above statement...
2. 9 marks] Strings. Consider the following definitions on strings Let U be the set of all strings Let s be a string. The length or size of a string, denoted Is, is the number of characters in s Let s be a string, and i e N such that 0 < ί < sl. We write s[i] to represent the character of s at index i, where indexing starts at 0 (so s 0] is the first character, and...
2. 9 marks] Strings. Consider the following definitions on strings Let U be the set of all strings. Let s be a string. The length or size of a string, denoted Is, is the number of characters in s Let s be a string, and ie N such that 0 Si< Is. We write si] to represent the character of s at index i, where indexing starts at 0 (so s(0 is the first character, and s|s -1 is the...
5. Symbolize the following argument and prove it is a valid argument. Let B ( x ) = x is a bear; D ( x ) = x is dangerous, and H ( x ) = x is hungry. Every bear that is hungry is dangerous. There is a hungry animal that is not dangerous. Therefore there is an animal that is not a bear. 6. In order to prove an quantificational argument invalid it is only necessary to find a...
#7. TRUE/FALSE. Determine the truth value of each sentence (no explanation required). ________(a) k in Z k2 + 9 = 0. ________(b) m, n in N, 5m 2n is in N. ________(c) x in R, if |x − 2| < 3, then |x| < 5. #8. For each statement, (i) write the statement in logical form with appropriate variables and quantifiers, (ii) write the negation in logical form, and (iii) write the negation in a clearly worded unambiguous English sentence....
9. Prove that the following kogical expressions aro logically equivalent by applying the law of logic 10. Give a logical expression with variables p, q, and r that's true only if p and q are false and r is true. 11. Predicates P and Q are defined below. The domain of discourse is the set of all positive integers. P(x): x is prime Qlx): x is a perfect square Are the following logical expressions propositions? If the answer is yes,...
4. H ere are some True/False questions. If your answer is "TRUE", there is no need to justify your answer. If your answer is "FALSE", then you should justity your answer with a counterexample or explanation. There are also some "short-answer" questions. . A. (True-False). Every simple field extension of K is a finite field extension. . B. (True-False). Let R⑥ F be a field extension. Suppose that F is a of u E F, and splitting field for the...
(d) Translate the following statement into predicate logic: “Every function f :R → R can be written as the sum of an even function and an odd function.” You can use the notation fi + f2 to represent the sum of functions fı and f2, and the notation f1 = f2 to represent the fact that fi and f2 are equal. 2n izo (e) Let n € N, and 20, 21, ..., Q2n E R. Let f: R + R...
2. Mark each statement True or False. Justify your answer. (6 points each) (a) (A\B) U ( BA) = (AUB) (ANB). (m) º (7.1+5) = (a :) n=2 1 3. Let A = {J,Q,K) and B = {4,0}. Find (list thc clements of) the set A x B. Find |A x B) and P(A x B) (9 points) 3