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Marysya12 [62]
3 years ago
8

Consider the following argument. If I get a Christmas bonus, I'll buy a stereo. If I sell my motorcycle, I'll buy a stereo. ∴ If

I get a Christmas bonus or I sell my motorcycle, then I'll buy a stereo. Let p = "if I get a Christmas bonus," q = "if I sell my motorcycle," and r = "I'll buy a stereo." Is the argument valid or invalid?
Select the answer that shows the symbolic form of the argument and justifies your conclusion. form:______.
p → r invalid, converse error
q → r
∴ p ∨ q → r
form:_____.
r → q valid, proof by division into cases
r → p
∴ r → p ∨ q
form:______.
r → q invalid, converse error
r → p
∴ r → p ∨ q
form:______.
p → r valid, proof by division into cases
q → r
∴ p ∨ q → r
form:_______.
r → q invalid, inverse error
r → p
∴ r → p ∨ q
Mathematics
1 answer:
Nuetrik [128]3 years ago
4 0

Answer:

p → r valid, proof by division into cases

q → r

∴ p ∨ q → r

Step-by-step explanation:

Let

p = "if I get a Christmas bonus,"

q = "if I sell my motorcycle,"

and

r = "I'll buy a stereo."

This can be written as:

If I get a Christmas bonus, I'll buy a stereo

p → r

If I sell my motorcycle, I'll buy a stereo

q → r

∴ If I get a Christmas bonus or I sell my motorcycle, then I'll buy a stereo.

∴ p ∨ q → r

To prove this argument we partition the argument into a group of smaller statements that together cover all of the original argument and then we prove each of the smaller statements. If you see the conclusion ∴ p ∨ q -> r so if the conclusion contains a conditional argument of form "If A1  or A2 or... or An then C ”, then we prove "If A1 then C", "If A2 then C" and so on upto "If An then C" . This depicts that the conclusion  C is true no matter which if the Ai holds true. This method is called proof by division into cases. In the given example, this takes the form:

p → r

q → r

p ∨ q

∴ r

Since proof by division into cases is an inference rule thus given argument is valid. Lets make a truth table to show if this argument is valid

p   q   r   p ∨ q   p → r    q → r    p ∨ q → r

0   0   0     0        1           1             1

0   0   1      0        0          0            0

0   1    0     1         1           0            0

0   1    1      1         0          1               1

1    0   0     1         0          1             0    

1    0   1      1         1           0            1    

1    1    0     1         0          0            0    

1    1    1      1         1           1              1    

An argument is valid if all of the premises are true, then the conclusion is true. So the truth table shows that the conclusion is true i.e. 1 where all premises are true i.e. 1. So the argument is valid.

Hence

p → r valid, proof by division into cases

q → r

∴ p ∨ q → r

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Dennis_Churaev [7]

Answer:

It would take approximately 6.50 second for the cannonball to strike the ground.

Step-by-step explanation:

Consider the provided function.

h(t)=-4.9t^2+30.5t+8.8

We need to find the time takes for the cannonball to strike the ground.

Substitute h(t) = 0 in above function.

-4.9t^2+30.5t+8.8=0

Multiply both sides by 10.

-49t^2+305t+88=0

For a quadratic equation of the form ax^2+bx+c=0 the solutions are: x_{1,\:2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}

Substitute a = -49, b = 305 and c=88

t=\frac{-305+\sqrt{305^2-4\left(-49\right)88}}{2\left(-49\right)}=-\frac{-305+\sqrt{110273}}{98}\\t = \frac{-305-\sqrt{305^2-4\left(-49\right)88}}{2\left(-49\right)}= \frac{305+\sqrt{110273}}{98}

Ignore the negative value of t as time can't be a negative number.

Thus,

t=\frac{305+\sqrt{110273}}{98}\approx6.50

Hence, it would take approximately 6.50 second for the cannonball to strike the ground.

6 0
3 years ago
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AfilCa [17]
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How many zeros does the quadratic function f(x) = x2 have
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A new neighborhood activity complex is being built in erie. The perimeter of the rectangular playing field is 256 yards. The len
maw [93]

Answer:

The length of the rectangular playing field is 84 yards and the width is 44 yards

Step-by-step explanation:

Let

x ------> the length of the rectangular playing field

y -----> the width of the rectangular playing field

we know that

The perimeter of the rectangular playing field is equal to

P=2(x+y)

P=256\ yd

so

256=2(x+y) ------> equation A

we have that

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3y=128+4

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Find the value of x

x=2(44)-4=84

therefore

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