The document is a problem set that contains 8 questions: 1. Name 3 elements and 3 subsets of the set A = {∅, a, b, c, d, e, f} and explain why a repeated element is irrelevant. 2. Find the union of the sets E = {2n : n is an integer} and O = {2n + 1 : n is an integer}. 3. Find the intersection of E and O. 4. Solve several linear equations. 5. Solve several quadratic equations. 6. Solve a cubic equation. 7. Generalize the distributive property to higher orders. 8. Pro

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.
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