Help Direct proof and indirect proof

[jonbinsdesu]

Baka may makasagot sa inyo mga ma'am/sir nahihirapan talaga ako pagdating sa math

 

[Deleted member 1371840]

Hi. Idk how to formally write proofs but I'll try to help
A.
1.
Representing the even numbers x and y:
x = 2p
y = 2q

x^2 + y^2
= (2p)^2 + (2q)^2
= 4p^2 + 4q^2
= 4(p^2 + q^2) = 2(2)(p^2 + q^2)

Having a factor of 2, we can conclude x^2 + y^2 is even.

2.
Representing the odd number x:
x = 2k + 1

x + x^2
= (2k + 1) + (2k + 1)^2
= 2k + 1 + (4k^2 + 2k + 1)
= 4k^2 + 4k + 2
= 2(2k^2 + 2k + 1)

Having a factor of 2, we can conclude x + x^2 is even.

 

[Deleted member 1371840]

B.
1.
Suppose that x and y are both even and x^2 + y^2 is odd. Since x and y are even, then
x = 2p
y = 2q
for some integers p and q.

Thus,
x^2 + y^2
= (2p)^2 + (2q)^2
= 4p^2 + 4q^2
= 4(p^2 + q^2) = 2(2)(p^2 + q^2)

Then the preceding integer 2(2)(p^2 + q^2) - 1 is odd since we assumed that x^2 + y^2 is odd.
This is impossible since there are no two consecutive integers that are both odd. Because the supposition that x^2 + y^2 is odd leads to a contradiction, we conclude that x^2 + y^2 is even.

 

[jonbinsdesu]

B.
1.
Suppose that x and y are both even and x^2 + y^2 is odd. Since x and y are even, then
x = 2p
y = 2q
for some integers p and q.

Thus,
x^2 + y^2
= (2p)^2 + (2q)^2
= 4p^2 + 4q^2
= 4(p^2 + q^2) = 2(2)(p^2 + q^2)

Then the preceding integer 2(2)(p^2 + q^2) - 1 is odd since we assumed that x^2 + y^2 is odd.
This is impossible since there are no two consecutive integers that are both odd. Because the supposition that x^2 + y^2 is odd leads to a contradiction, we conclude that x^2 + y^2 is even.

Salamat ng marami sir naguguluhan talaga ako sa lessons di talaga ako maka catch up sa explanations po

 

[Deleted member 1371840]

You're welcome! Btw I'm a girl. Haha

What subject is this? Discrete math, elem analysis or number theory? I didn't encounter problems like this while in college, that's why I don't know how to write a formal proof. So please check mo na lang din.

Just post here whenever you need help. Good luck on your studies TS!

 

[jonbinsdesu]

You're welcome! Btw I'm a girl. Haha

What subject is this? Discrete math, elem analysis or number theory? I didn't encounter problems like this while in college, that's why I don't know how to write a formal proof. So please check mo na lang din.

Just post here whenever you need help. Good luck on your studies TS!

Di ko rin po sure ma'am, Mathematics in the Modern World po pala sub namin feel ko belong po siya sa discreet po

 

[jonbinsdesu]

Uhm by the way po pwede po ba ako maka ask sa B#2 ipapacheck ko lang po if tama po ang answer contrapositive po ang need kasi naguguluhan po ako

 

[jonbinsdesu]

 

[zero_o_clock]

Hello. This is the only "attack" I can think of. I cannot think of any other way this can be proved.