Originally Posted by innuenn
1.) (For every n in the element natural numbers): If n>4 and n is a perfect square, then n-1 is not a prime number.
2.) (For every n in the element natural numbers): If n is not divisible by 3, then n^2+2 is divisible by 3.
3.) If n is in the element of natural number and n is a perfect square, then the units digit of n is not 2.
4.) If a,b are in the element of natural numbers and both a and b are odd, then x^2+ax+b can not be factored into a product of two linear factors ( A linear factor has the form cx+d, where c,d an element of natural numbers).
5.) Prove by induction: 1+x+x^2+...+x^n=(x^(n+1)-1)/(x-1)
6.)
7.) Prove (3^n-1)/2 = (Sigma) 3^(i-1)
8.) Define a sequence of numbers {An} by An=2A(n-1)+A(n-2), where A1=5, A2=10. Prove for n is an element of natural numbers with n(Greater or equal to)3 that An(less then)3^n
9.) For n in the element of integers, let Cn=[n,n+1) and let e={Cn: n is in the element of integers}. Find (union)e and find (intersection)e.
10.) Prove or give a counter example for each statement:
A.) If A(union)C (subset of) B(union)C, then A(subset of)B
B.) If A(Intersects)C (subset of) B(Intersects)C, then A(subset of)B
C.) P(A)-p(B) (subset of) p(A-B)
11.) Give an example of a nested family {Ai: I is in the element of natural numbers} for each condition below:
A .) (Intersection)Ai = (3,5]
B.) (Intersection) Ai = {3,5}
12.) Let Sbe the set of all 2nd degree polynomials with reel roots i.e. S={ax^2+bx+c: a,b,c in the element of real numbers and b^2-4ac(greater or equal to)0} Define a relation R on S by fRg iff f and g have the exact same zeros. Answer the following:
A.) Prove R is an equivalence relation on S (i.e. show R is symmetric, reflexive and transitive)
B.) Find [x^2-1] (i.e. find the equivalence class x^2-1)
