Western Governors University Foundations of Mathematics Worksheet I try to work 2 times but the feedback is not good, I only get 4/10 points. please help

Western Governors University Foundations of Mathematics Worksheet I try to work 2 times but the feedback is not good, I only get 4/10 points.

please help me work it

it likes first times problem. need formal proof.

I need pdf file, you can use latex, word or any way you work convenience

and you need pay attention for the writing Mathematics 403
Foundations of Mathematics
Spring 2020
Portfolio Project
Problem 5
Let A be a finite set with n elements. Prove that
X
2card(X) = 3n .
X?P(A)
Portfolio 6
Proof:
We prove by induction:
3
1
n = 0, card(A) = 0, A = ?, P(A) = {?}
2card(X) = 2card(?) = 20 = 1 = 30 = 3n 2
The claim hold for n = 0
Now suppose the claim hold for n = k ? N ?
When n = k + 1, A is a set with n = k + 1 elements
4
and denote A = {x1 , x2 , · · ·, xk , xk+1 }
def B = {x1 , x2 , · · ·, xk },thus A = B ? {xk+1 }
card A = k + 1 = n, card B = k
W
Phen
X?P(A)
5
?X ? P(A):
1)xk+1 ? X, X/{xk+1 }6 ? B
2)xk+1 ?
/ X, X ? B
on the contrary
?Y ? P(B):
1)Y ? P(A)
2)(Y ? {xk+1 }) ? P(A)
Thus def Q(B) = {X ? {xk+1 } : X ? P(B)}
We note P(A) = P(B) ? Q(B),P(A) ? Q(B) = ?
Thus
P
P
P
card(X)
= X?P(B) 2card(X) + X?Q(B) 2card(X)
(assumption)
X?P(A) 2
P
k
card(Y ?{xk+1 })
= 3 + Y ?P(B) 2
(card(Y ) ? {xn+1 } = card(Y + 1))
P
k
card(Y )+1
= 3 + Y ?P(B) 2
P
= 3k + 2 Y ?P(B) 2card(Y )
(assumption)
k
k
=3 +2·3
= 3k+1
= 3n
Thus the claim hold for n = k + 1
By induction we finish the proof.
1
Comment Summary
Page 1
1. card shouldn’t be italic.
2. Your proof should have a normal sentence and paragraph structure. Lines should run from margin to
margin—not stop in the middle of the page.
3. Your proof should be written in complete sentences. It shouldn’t have strings of unrelated formulas.
4. “define”?
5. Formal proofs shouldn’t contain logical symbols.
6. What do these lines mean?

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