![Brennen435](/avatars/26890.jpg)
Brennen435
21.02.2020 •
Mathematics
Determine whether each of these compoundpropositions is satisfiable. a) (p∨q∨¬r)∧(p∨¬q∨¬s)∧(p∨¬r∨¬s)∧ (¬p ∨¬q∨¬s)∧(p∨q∨¬s) b) (¬p∨¬q∨r)∧(¬p∨q∨¬s)∧(p∨¬q∨¬s)∧ (¬p ∨¬r ∨¬s)∧(p∨q∨¬r)∧(p∨¬r∨¬s) c) (p∨q∨r)∧(p∨¬q∨¬s)∧(q∨¬r∨s) ∧(¬p∨ r ∨s) ∧(¬p∨q∨¬s)∧(p∨¬q∨¬r) ∧(¬p∨ ¬q ∨s)∧(¬p∨¬r∨¬s)
Solved
Show answers
More tips
- B Business and Finance How to Create a Business Plan? Your Ultimate Guide...
- F Food and Cooking Deflope: What is it and how does it work?...
- F Food and Cooking Why Doesn t the Confirmation Link Come to Email?...
- F Food and Cooking How to Get Reconfirmation of Registration?...
- S Science and Technology Discovering the Anatomy of an LCD TV Screen...
- H Health and Medicine What You Need to Know About Nasal Congestion in Infants: Causes, Symptoms, and Treatment...
- A Auto and Moto What is the Average Lifespan of an Engine in a Car?...
- C Computers and Internet Make Money Online: Secrets and Essential Ways...
- A Auto and Moto How Can Parking Sensors Help Drivers?...
- S Style and Beauty How to Properly Apply Eye Makeup: Tips from a Professional Makeup Artist...
Answers on questions: Mathematics
- M Mathematics Anumeha is mowing lawns for a summer job. for every mowing job, she charges an initial fee of \$10$10dollar sign, 10 plus a constant fee for each hour of work. her fee for a 555-hour...
- M Mathematics Yes or no? the integral of a to b f(x)dx is a number...
- M Mathematics A daycare employee earns $\$10.90$ per hour for the first $40$ hours he works in $1$ week. He is paid $1.5$ times that rate for each additional hour that he works. How much money...
- C Computers and Technology Select the correct answer. Troy, an aspiring screenwriter, aspires to work with a famous director known for his philanthropic work. Troy gets an appointment with the director to...
- P Physics What is the kinetic energy of a 7,500 n automobile which is moving at 55 m/s?...
- M Mathematics Démontrer que x²-4x+5=(x-2)²+1 ; en déduire que x²-4x+5=0 n a pas de solution...
Ответ:
See below
Step-by-step explanation:
A compound proposition is satisfiable if there exists a combination of truth values of the component propositions that make the compound proposition true.
A compound conjunction p1∧p2∧p3∧...∧pn is true iff pi is true for all i.
A compound disjunction p1∨p2∨p3∨...∨pn is true iff pi is true for some i.
a) (p∨q∨¬r)∧(p∨¬q∨¬s)∧(p∨¬r∨¬s)∧(¬p∨¬q∨¬s)∧(p∨q∨¬s) is satisfiable.
Assume that p is true, q is true, r is false and s is false.
Since p is true, then (p∨q∨¬r), (p∨¬q∨¬s) and (p∨¬r∨¬s) are true.
Now, s is false, then ¬s is true. Then (¬p∨¬q∨¬s) and (p∨q∨¬s) are true.
Combine both statements to obtain that (p∨q∨¬r)∧(p∨¬q∨¬s)∧(p∨¬r∨¬s)∧(¬p∨¬q∨¬s)∧(p∨q∨¬s) is true.
b) (¬p∨¬q∨r)∧(¬p∨q∨¬s)∧(p∨¬q∨¬s)∧(¬p∨¬r∨¬s)∧(p∨q∨¬r)∧(p∨¬r∨¬s) is satisfiable
Assume that p is false, q is false, r is false and s is false.
Since p is false, ¬p is true, then (¬p∨¬q∨r), (¬p∨q∨¬s) and (¬p∨¬r∨¬s) are true.
Since s is false, ¬s is true, then (p∨¬q∨¬s) and (p∨¬r∨¬s) are true.
Since s is false, ¬s is true, then (p∨q∨¬r) is true.
Combining all of these, the compound proposition is satisfiable (each component of the disjunction is true).
c) (p∨q∨r)∧(p∨¬q∨¬s)∧(q∨¬r∨s)∧(¬p∨r∨s)∧(¬p∨q∨¬s)∧(p∨¬q∨¬r)∧(¬p∨¬q ∨s)∧(¬p∨¬r∨¬s) is satisfiable.
Assume that p is true, q is true, s is true and r is false.
Since p is true, (p∨q∨r), (p∨¬q∨¬s) and (p∨¬q∨¬r) are true.
Since s is true, (q∨¬r∨s), (¬p∨r∨s) and (¬p∨¬q ∨s) are true.
Since ¬r is true, (p∨¬q∨¬r) and (¬p∨¬r∨¬s) are true.
Since q is true, (¬p∨q∨¬s) is true.
Combining all of the above, the compound proposition is satisfiable.
Remark: there are unsatisfiable propositions, but proving that a proposition is unsatisfiable requires a general argument. For example, (p∧q)∧(¬p) is unsatisfiable, since it contains the contradiction (p∧¬p).
Ответ:
3 + 7 + 2= 12
Step-by-step explanation:
It is made true because the equation adds up to the number. If we take 3+7 it equals to 10. Then we take 10 and add it to 2, which gives us 12.
Hope this helps :)