1. Consider the following hypotheses:H1 : ∃x (p(x) ∧ q(x)) H2 : ∀x (q(x) → r(x))
Use rules of inference to prove that the following conclusion follows from these hypotheses:
C : ∃x (p(x) ∧ r(x))
Clearly label the inference rules used at every step of your proof.
2. Consider the following hypotheses:
H1 : ∀x (¬C(x) → ¬A(x)) H2 : ∀x (A(x) → ∀y B(y)) H3 : ∃x A(x)
Use rules of inference to prove that the following conclusion follows from these hypotheses:
C : ∃x (B(x) ∧ C(x))
Clearly label the inference rules used at every step of your proof.
3. Consider the following predicate quantified formula:
∃x ∀y (P (x, y) ↔ ¬P (y, y))
Prove the unsatisfiability of this formula using rules of inference.
Answers
Answer:
See deductions below
Step-by-step explanation:
1)
a) p(y)∧q(y) for some y (Existencial instantiation to H1)
b) q(y) for some y (Simplification of a))
c) q(y) → r(y) for all y (Universal instatiation to H2)
d) r(y) for some y (Modus Ponens using b and c)
e) p(y) for some y (Simplification of a)
f) p(y)∧r(y) for some y (Conjunction of d) and e))
g) ∃x (p(x) ∧ r(x)) (Existencial generalization of f)
2)
a) ¬C(x) → ¬A(x) for all x (Universal instatiation of H1)
b) A(x) for some x (Existencial instatiation of H3)
c) ¬(¬C(x)) for some x (Modus Tollens using a and b)
d) C(x) for some x (Double negation of c)
e) A(x) → ∀y B(y) for all x (Universal instantiation of H2)
f) ∀y B(y) (Modus ponens using b and e)
g) B(y) for all y (Universal instantiation of f)
h) B(x)∧C(x) for some x (Conjunction of g and d, selecting y=x on g)
i) ∃x (B(x) ∧ C(x)) (Existencial generalization of h)
3) We will prove that this formula leads to a contradiction.
a) ∀y (P (x, y) ↔ ¬P (y, y)) for some x (Existencial instatiation of hypothesis)
b) P (x, y) ↔ ¬P (y, y) for some x, and for all y (Universal instantiation of a)
c) P (x, x) ↔ ¬P (x, x) (Take y=x in b)
But c) is a contradiction (for example, using truth tables). Hence the formula is not satisfiable.
Related Questions
Your office is raising money for charity by collecting aluminium cans for recycling. They get 36p per kilo of cans. A kilo is approximately 72 cans. They have collected 8645 cans How much money have they raised?
Is it true that Confidence intervals are always close to their true population values?
In Exercise 5.21, we learned that a Rasmussen Reports survey of 1,000 USadults found that 42% believe raising the minimum wage will help the economy. Construct a 99% confidence interval for the true proportion of US adults who believe this.
What is 231 scaled down by a factor of 1/10
Answers
Answer:
21.45 miles
Step-by-step explanation:
Key:
t=time
d=distance
We need to find the time the lion ran first.
The time the lion ran:
13.75=50•t
13.75=50t
-50t=-13.75
t=-13.75•-1/50
t = 0.275
So the lion ran in 13.75 miles in 0.275 of an hour.
Now, we need to figure out the cheetah.
0.275•78=21.45 miles
I hope this helps :-)
Answers
Answer:
491400
Step-by-step explanation:
Given : A club has 28 members.
To Find : . How many ways are there to choose four members of the club to serve on an executive committee?
Solution:
We are supposed to choose four members of the club out of 28.
So, we will use combination
Formula :
n = 28
r = 4
Substitute the values :
Hence there are 491400 ways o choose four members of the club to serve on an executive committee
2. 2 units down and 3 units up
3. 2 units right and 3 units up
4 2 units left and 3 units right
Answers
Answer:
i believe it is 4 but i'm not so sure
Step-by-step explanation:
:)
Final answer:
The graph of g(x)=(x-2)^2+3 compared to the graph of f(x)=x^2 is translated 2 units to the right and 3 units upwards.
Explanation:
To understand the transformation of graphs in mathematical terms, consider the initial function f(x) = x^2. The transition to the new function g(x)=(x-2)^2+3 is a result of a shift or translation of the graph. This transformation behaves as per the following rule: g(x) = f(x-h)+k where 'h' units is the horizontal displacement and 'k' units is the vertical displacement.
In the function g(x), x shifts two units to the right (as indicated by (x-2)) and three units upward (as indicated by +3).
So, the correct answer is 2 units right and 3 units up.
Learn more about graph transformation here:
#SPJ3
Answers
Answer:
y = -7/5x - 6.7
Step-by-step
y = -7/5x + 6. A Line that is parallel will always have the same slope. The slope is m. m in this situation = -7/5x. A lines equation is y = mx + b. m = -7/5. y = -7/5x + b. Now to find b we can substitute the given point which goes through the new line, (2, -6). In this point x =2 and y = -6. Now substitute the x and y values into our equation. y = -7/5x + b is now -6 = -7/5(2) + b. -7/5(2) = -7/10. -6 = 7/10 + b. Subtract 7/10 from -6. Its -6 and 7/10 or -6.7 . -6.7 = b. b = -6.7. Now substitute the b value into the equation. y = -7/5x -6.7.
Answers
Answer:
A={5,10,15,20,25,30}
Step-by-step explanation:
Theres no explanation
Answers
Answer:
After 4 years working for the company, he would make $79,000 of salary.
His salary after t years will be:
[te]S(t) = 69000 + 2500t[/tex]
Step-by-step explanation:
David just accepted a job at a new company where he will make an annual salary of $69000. David was told that for each year he stays with the company, he will be given a salary raise of $2500.
This means that after t years, his salary will be given by:
[te]S(t) = 69000 + 2500t[/tex]
How much would David make as a salary after 4 years working for the company?
[te]S(4) = 69000 + 2500*4 = 69000 + 10000 = 79000[/tex]
After 4 years working for the company, he would make $79,000 of salary.