Question

8. Prove that if n is a perfect square, then n + 2 is not a perfect square. 9. Use a proof by contradiction to prove that the sum of an irrational number and a rational number is irrational 17. Show that if n is an integer and n3 + 5 is odd, then n is even using a) a proof by contraposition. b) a proof by contradiction. 42. Prove that these four statements about the integer n are equivalent: (i) na is odd, (ii) 1-n is even, (ii) nis odd, (iv) 2 + 1 is even. 1. Prove that 12 + 1 > 2" when n is a positive integer with 1<n<4. 34. Prove that / is irrational.

Please send me the solutions for the above 6 questions. Please send it as tomorrow I have an exam. Thank You. ?

Answer 1

6 (v =)) we will prove its contraposition. let is even then there is an integer k such that n= 2k. Next, we have n3 m² + 1 = UK²+1 - 212²)+1 which means that that there is an integer m = 2k² such that m² + 1 = rm ti or m² + 1 is odd. civeu) let an is odd, then there is an. integer k such that n = 2k +1. Next, we have in 2 + 1 = 4k tuk titi = 2(24²+2kt) which means that there is an integer m= 2K²+2kt such that m²+1=2m. se n²+1 is even.

& If Prove that if n is a perfect square, then M+2 is not a perfect square. fisst for mal nil, real, n+2= 3. 3 is not a perfect square for no, There are no perfect square between n and (n+1)2, exclusive. for 7,2 n2 n² to 4 (n+1) 2 so ntr is not a perfect square.

6 use a proof by contradiction to prove that the sum of an irrational number and rational number is irrational. Given that r is rational number, and a is an irrational number Since ris rational, as is also rational, thus the sum of sto andas must be a rational number [ sum of two rational mo is rational Thus (r+a)+(-8) -n is rational This contradicts our original hypothesis that is irrational. Therefore it must not be true that sta is rational de &tx must be irrational

1 show that if n is an integer and n²+5 is odd, then nis enon using a) a proof by contraposition. a proof by contradiction. 5) « Phot, Сое as Proof by contraposition The contraposition of the statement is "If n is odd then nuts is events Hence, to proof the contraposition, we need to assume that n is odd. By the definition of odd number, there is an integer & such that naakt n= akt into th substituting we get n3+5 = (2x + 1)² +5

- [8 kº +12 k? + 6k +1) + 5 = 8K² +12 K² + 6x + 6 = 2 (4x²+6K² +3 K+3) Thus, we can find an integer e. such that e= 4K²+6K²+36 +3 then nits=20 That means, nits is even. Since the contraposition is true then. then the original statement is also true

8 Proof by contradiction Suppose that nots is odd and is not even. Because n is not even then & is odd. By the definition of odd numbers, there is an integer K such that Me2k + 1 Substituting v= 2Kt into its we get (nts ) = (2x H3 +5 =(8K² H2k² +6K+1) +5 = 843 +124² +6k to = 2(4K to k² + 3k+3) Thus, we can find an integer e = 4x3 +68² +34 +3 such that n375 =21

That means to is even So far , our assumption nih odd leads us to contradiction that uts is both even and odd. This is contradiction. Hence & is odd must be a false statement. This completes the proof that must be an even number.

1 statements about equivalent Prove that these four the integer n are is n² is odd. nylon is even lii) ne is odd in n2t1 is even Sol Owe set a new statement v nis odd. ( u) we will prove it contraposition let is even then there is an integer k such that n=2k. Next, we have n² = (2x)2 = 46² = 2(24²) which means that there is an integer m= 2k² such that n² = 2m, or m² is enen.

(Ev) let nes odd then there is an Integer & such that n=2kt. Next, we have m? - (2x +132 = 4K +91 + =2(2k% +2k)+1; which means that there is an integer m= 26² +2k such that n² = 2mt, or 2 is odd.

il v) we will prove its contraposition cet nis cum then there is an integer K such that n=2k. Next, we have I-n= 1-12k) = -2k t1 = 2(-k)+1, which means that there is an integer m=uk such that no 1-n=2m +1 or In is odd (iiku) ut nis odd, there is an integer e such that n=24+. Next, we have i-nz 1- (2k +1)=- 2K = 26-) which means that there is an integer m= -K such that 1-naam. or i-n is enen.

② chiu) we will prove its contraposition. let mis even then there is an integer k such that naak, Next all have n3 = (2x)3 = 843 – 2 (443) which means that there is an integer m= 443 such that n3 = 2m, or n3 is even. ( Ev) let in is odd, then there is an integer k such that m= 2kt Next, ull have na (2k +133 = 843 + 84² +4 kth = 2(46° + 4*? +2k +1 which means that there is an integer mauk² tuk² such that n3 = 2m +1 or ne is odd.