1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
BlackZzzverrR [31]
2 years ago
12

Prove the formula that:

Mathematics
1 answer:
azamat2 years ago
5 0

Step-by-step explanation:

Given: [∀x(L(x) → A(x))] →

[∀x(L(x) ∧ ∃y(L(y) ∧ H(x, y)) → ∃y(A(y) ∧ H(x, y)))]

To prove, we shall follow a proof by contradiction. We shall include the negation of the conclusion for

arguments. Since with just premise, deriving the conclusion is not possible, we have chosen this proof

technique.

Consider ∀x(L(x) → A(x)) ∧ ¬[∀x(L(x) ∧ ∃y(L(y) ∧ H(x, y)) → ∃y(A(y) ∧ H(x, y)))]

We need to show that the above expression is unsatisfiable (False).

¬[∀x(L(x) ∧ ∃y(L(y) ∧ H(x, y)) → ∃y(A(y) ∧ H(x, y)))]

∃x¬((L(x) ∧ ∃y(L(y) ∧ H(x, y))) → ∃y(A(y) ∧ H(x, y)))

∃x((L(x) ∧ ∃y(L(y) ∧ H(x, y))) ∧ ¬(∃y(A(y) ∧ H(x, y))))

E.I with respect to x,

(L(a) ∧ ∃y(L(y) ∧ H(a, y))) ∧ ¬(∃y(A(y) ∧ H(a, y))), for some a

(L(a) ∧ ∃y(L(y) ∧ H(a, y))) ∧ (∀y(¬A(y) ∧ ¬H(a, y)))

E.I with respect to y,

(L(a) ∧ (L(b) ∧ H(a, b))) ∧ (∀y(¬A(y) ∧ ¬H(a, y))), for some b

U.I with respect to y,

(L(a) ∧ (L(b) ∧ H(a, b)) ∧ (¬A(b) ∧ ¬H(a, b))), for any b

Since P ∧ Q is P, drop L(a) from the above expression.

(L(b) ∧ H(a, b)) ∧ (¬A(b) ∧ ¬H(a, b))), for any b

Apply distribution

(L(b) ∧ H(a, b) ∧ ¬A(b)) ∨ (L(b) ∧ H(a, b) ∧ ¬H(a, b))

Note: P ∧ ¬P is false. P ∧ f alse is P. Therefore, the above expression is simplified to

(L(b) ∧ H(a, b) ∧ ¬A(b))

U.I of ∀x(L(x) → A(x)) gives L(b) → A(b). The contrapositive of this is ¬A(b) → ¬L(b). Replace

¬A(b) in the above expression with ¬L(b). Thus, we get,

(L(b) ∧ H(a, b) ∧ ¬L(b)), this is again false.

This shows that our assumption that the conclusion is false is wrong. Therefore, the conclusion follows

from the premise.

15

You might be interested in
A copper key is placed in a cylindrical container with a 3-cm diameter. The water level rises from 14 cm to 14.6 cm. What is the
Tomtit [17]

Answer:

The volume is about 4.24 cm^3.

Step-by-step explanation:

From the question we are told that:

Diameterd= 3-cm

Radius r=d/2=>1.5cm

Height risen14.6-14=0.6cm

Generally the equation for Volume of a Key is mathematically given by

Since Volume of a cylinder is Given as

V=\pir^2h

Therefore

V=\pi*1.5^2*0.6

V=4.2417\\\\V=4.24 cm^3

Correct Option A

The volume is about 4.24 cm^3.

3 0
3 years ago
a stone falls from a ledge and takes 8 seconds to the hit the ground. the stone has an original velocity of 0 m/s. how tall is t
Verizon [17]
Using equation of motion.

S = ut + (1/2)gt²

Where S = Height, t = time, g = acceleration due to gravity = 9.8 m/s²

time, t = 8 seconds 

Since it was dropped from height, initial velocity, u = 0

S = 0*t + (1/2)gt²

S = (1/2)gt²

S = (1/2)*9.8*8²  = 4.9 * 8 * 8 =  313.6

Therefore the ledge is 313.6 m tall.
6 0
3 years ago
Read 2 more answers
20<br> X<br> 40°<br> solve for x
kirza4 [7]
Could you please formulate you answer better? So we can get a bette understanding
8 0
3 years ago
Find all solutions of the equation. (enter your answers as a comma-separated list.) x2 + 5x + 9 = 0
erik [133]
X,=-(5±✓25-36) ,/ 2

x={-5±✓-11)/2

x= -5/2 ± i✓11/2
4 0
3 years ago
Gor is running a bakery. He has made a street sign to direct people to his
lys-0071 [83]

Answer:

B.

Step-by-step explanation:

The street sign is composed of two rectangles and 1 triangle.

Painted area = area of triangle + area of rectangle 1 + area of square

✔️Area of triangle = ½bh

b = 15 in.

h = 8 in.

Area of triangle = ½*15*8 = 60 in.²

✔️Area of rectangle 1 = L*W

L = 50 in.

W = 6 in.

Area = 50*6 = 300 in.²

✔️Area of Square = s²

s = 15 in.

Area = 15² = 255 in.²

✅Painted area = 60 + 300 + 225 = 585 in.²

7 0
2 years ago
Other questions:
  • Find the value of |4|
    15·1 answer
  • Richard wants to join a gym. Dynamic Gym has a membership fee of $50 and costs $35 each month. Cheater Fitness has a membership
    9·1 answer
  • IMAGE BELOW The equations x minus 2 y = 4, 4 x + 5 y = 8, 6 x minus 5 y = 15, and x + 2 y = 0 are shown on the graph below.
    5·1 answer
  • what solid 3D object is produced by rotating the triangle about line m with a height of 8 and radius 4
    6·1 answer
  • A pair of socks priced at 2.70 was on sale for 10% off. What was the original price of the socks
    8·1 answer
  • What is the measure of AngleM?<br><br> °
    9·2 answers
  • Please help meeeeeeeeeeeew
    12·2 answers
  • Which table represents a linear function?
    12·1 answer
  • 16.
    13·1 answer
  • If y = 3x + 1 is changed to y = 4x + 1, how would the graph of the new function compare with
    15·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!