Give proof for the following statements. In writing your proofs, be sure to mimic the method shown...

Question

Give proof for the following statements. In writing your proofs, be sure to mimic the method shown in the lesson. Effort should be taken now to write mathematics that is acceptable to those reading your work. 1) If x is an integer divisible by four, then x is the difference of two squares. 2) If two onto functions can be composed then their composition is onto. (Recall a function f: X?Y is onto if for every a in Y there is an element b in X for which f(b) = a.) 3) The square root of 2 is irrational. 4) There are an infinitely many primes. 5) The product ab of integers a and b is even if and only if at least one of the integers is even. 6) For all positive integers n, 1^2 + 2^2 + … + n^2 = (n)(n=1)(2n+1)/6.

Document Preview:

Assignment: Proof Give proof for the following statements. In writing your proofs, be sure to mimic the method shown in the lesson. Effort should be taken now to write mathematics that is acceptable to those reading your work. If x is an integer divisible by four, then x is the difference of two squares. If two onto functions can be composed then their composition is onto. (Recall a function f: X?Y is onto if for every a in Y there is an element b in X for which f(b) = a.) The square root of 2 is irrational. There are an infinitely many primes. The product ab of integers a and b is even if and only if at least one of the integers is even. For all positive integers n, 1^2 + 2^2 + … + n^2 = (n)(n=1)(2n+1)/6.

Assignment: Proof Give proof for the following statements. In writing your proofs, be sure to mimic the method shown in the lesson. Effort should be taken now to write mathematics that is acceptable to those reading your work. 1) If x is an integer divisible by four, then x is the difference of two squares. 2) If two onto functions can be composed then their composition is onto. (Recall a function f: X(Y is onto if for every a in Y there is an element b in X for which f(b) = a.) 3) The square root of 2 is irrational. 4) There are an infinitely many primes. 5) The product ab of integers a and b is even if and only if at least one of the integers is even. 6) For all positive integers n, 1^2 + 2^2 + … + n^2 = (n)(n=1)(2n+1)/6.

005_xf1o8pk-3et5uziw.doc

David · Accepted Answer

1)  
   Let x be an integer divisible by 4 . 
So, x = 4n  ;   n = an integer
  4n = 2 * 2n = ((n+1) + (n-1)) x ((n+1) - (n-1)) = (n+1)
2
 - (n-1)
2

so, if x is an integer divisible by 4 , we can write it as a difference of squares of two different integers .
2)   
   let f & g are two onto functions . 
i.e for every a in the domain of f , there exist a value b in the range of f such that f(a) = b ; 
& for every s in domain of g , there exist a vlue t in the range of g , such that g(s) = t ;
if we can compose f & g then , 
                                   i) h(x) = f(g(x))  
                                  ii) k(x) = g(f(x))
i) h(x) = f(g(x))
 since we can compose f on g  ,   range of g should be a subset of domain of f . so ,

Give proof for the following statements. In writing your proofs, be sure to mimic the method shown in the lesson. Effort should be taken now to write mathematics that is acceptable to those reading...

Answer To: Give proof for the following statements. In writing your proofs, be sure to mimic the method shown...

Answer To This Question Is Available To Download

Related Questions & Answers

Submit New Assignment