21. A conditional statement is logically equivalent toitsA. InverseB. ContrapositiveC. ConverseD. None22. What is the equivalent contrapositive of "If it rains,then they cancel school"?A. If they do not cancel school, then it does not rain.B. If it rains, then they do not cancel school.C. If it does not rain, then they cancel school.D. No equivalent contrapositive23. "The chair in the living room is red. The chair in thedining room is red. The chair in the bedroom is red. Allthe chairs in the house are red." These statementsillustrateI. Inductive Reasoning II. Deductive ReasoningA. I onlyB. Il onlyC. Both I and IID. Neither I nor I|

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Description: 21. A conditional statement is logically equivalent toitsA. InverseB. ContrapositiveC. ConverseD. None22. What is the equivalent contrapositive of "If it rains,then they cancel school"?A. If they do not cancel school, then it does not rain.B. If it

Answered: 21. A conditional statement is…

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21. A conditional statement is logically equivalent to its A. Inverse B. Contrapositive C. Converse D. None

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

ChapterP: Preliminary Concepts

SectionP.CT: Test

Problem 1CT

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Translate the given statement into symbolic form. Be sure to define all pred- icates used. Use the variable a with domain D = {a : a is an animal } as your main variable. Then, negate the symbolic statement. In your answer, negations should appear only in front of predicates. Finally, rewrite the negated statement clearly in plain English. "Any animal that is wild can be tamed." 2.
Write each of the following conditional statements and the corresponding converse, inverse, and contrapositive as English sentences in the form “if ?, then ?.”Example: Whenever it rains, it pours. Answer: Original statement: If it rains, then it pours. Converse: If it pours, then it rains. Inverse: If it doesn’t rain, then it doesn’t pour. Contrapositive: If it doesn’t pour, then it doesn’t rain. 1.1 You can do it if you put your mind to it. 1.2 Being cold implies that you need a jacket. 1.3 Petting the cat is sufficient for it to purr.
ii. Let F(x,y) be the open sentence "x can fool y" where the domain of discourse is the set of all people in the world. Express each of these statements using logical quantifiers. c. Everybody can fool somebody. e. Everyone can be fooled by somebody. f. There is somebody who can be fooled by everybody.
Express this in predicate logic: Each email address has exactly one email box. M(x): x is an email address B(x,y): x has an email box y
Check all the statements that are true: A. It is possible to define disjunction using only negations and conjunction. B. Domain-restricted universal quantification is logically equivalent to the unrestricted universal quantification of a conditional. C. (p → a) v (p → →q) is a tautology. D. The negation of a conjunction is the disjunction of the negations. E. Inverse and converse are logically equivalent. F. A conditional is false when both premise and conclusion are false. G. The contrapositive is the inverse of the converse. H. A predicate is a proposition-valued function. I. Domain-restricted existential quantification is logically equivalent to the unrestricted existential quantification of a conditional. J. (pq) ^ (pq) is a contradiction.
21. Let the following predicates be given. The domain is all computer science classes. I(r) = "x is interesting." U(x) = "x is useful." H(x, y) = “x is harder than y." M(x, y) = “x has more students than y." (a) Write the following statements in predicate logic. i. All interesting CS classes are useful. ii. There are some useful CS classes that are not interesting. iii. Every interesting CS class has more students than any non- interesting CS class. (b) Write the following predicate logic statement in everyday English. Don't just give a word-for-word translation; your sentence should make sense. (3)[I(x) A (Vy)(H(x,y)→ M(y, x))] (c) Formally negate the statement from part (b). Simplify your negation so that no quantifier lies within the scope of a negation. State which derivation rules you are using. (d) Give a translation of your negated statement in everyday English.
Determine whether the logical statement is a tautology, a contradiction or neither: (p→q) → (p v q). [Hint: Use a truth table with 4 rows; you need to check the last row; all T is a tautology: all F is a contradition; a mixture of T and F is neither.] Example statement: If adopting new software leads to savings, then we'll either adopt new software or save money. The parentheses fit in the statement like this: If (adopting new software leads to savings), then (we'll either adopt new software or save money). This interpretation gives adopting new software leads to savings = If we adopt new software, then we'll save money O a. Tautology O b. Neither Oc. Contradiction