Problem Set 1

Problem Set 1

• 1.
  Problem Set 1 March 23, 2011 1. A = ∅, a, b, c, d , e, f where ∅ is said to be the empty set (the set which con- tains nothing). A fundamental theorem in abstract algebra is that all sets con- tain the empty set, or notationally; ∀A ∅ ⊂ A. Name 3 elements in A using appropriate notation. Name 3 distinct subsets of A. Why must we say that a repeated element in a set is irrelevant (that is, {a} = {a, a})? Hint: Do the sub- sets of A contain ∅? 2. Let E = {..., −2, 0, 2, 4, 6...} = {2n : ∀n Z} and O = {2n + 1 : ∀n Z}. What then is E ∪ O? 3. The intersection of two sets A and B is defined to be a third set C , such that C contains only elements of that exist in both A and B.We use the notation, C = A ∩ B . What is E ∩ O? 4. Solve the following linear equations (first order polynomials): (a) x + 5 = 0 (b) x − 2 = 3 (c) 1 x = π 2 5. Solve the following quadratic equations: (a) x 2 = 4 (b) x 2 − 10x + 25 = 0 (c) x 2 − 12x = −36 (d) x 2 + 6x + 9 = 0 (e) x 2 = −49 − 15x 6. Solve the cubic equation: x 3 + 7x 2 + 15x + 9 = 0 7. In the first lesson I generalized the distributive property of Algebra to binomi- als; (a + b)(d + e) = ad + bd + ae + be. Can you generalize this to n-orders? In other words, how else can you express... m n S= αi βj i =1 j =1 b 2 8. Prove the quadratic formula; x = −b± 2a −4ac . Hint: Start with the general quadratic 2 polynomial ax + bx + c = 0. Look up the method of “completing the square” for solving quadratics. 1