Math 2510 Test II
November 25, 2008 Show your work. 1. (2 points each) (a) State the Continuum Hypothesis. (b) What is Well-Ordering Property of the natural numbers. (c) State the triangle inequality. (d) If S R, what is the definition of inf S? (e) For x
Math 2510: Real Analysis I Homework Solutions 13
Fall 2008
13 Topology of the Reals
13.1 Let S R Mark each statement True or False. Justify each answer.
(a) int S bd S =
TRUE: Suppose that y is in int S . Then there is an > 0 so that N (y, ) S . But then
Math 2510: Real Analysis I Homework Solutions 12
Fall 2008
12 The Completeness Axiom
12.3 For each subset of R, give its supremum and its maximum, if they exist. Otherwise, write none.
(a) cfw_1, 3
supcfw_1, 3 = maxcfw_1, 3 = 3
(c) [0, 4]
sup[0, 4] = max[
Math 2510: Real Analysis I Homework Solutions 7
Fall 2008
11 Ordered Fields
11.3b (x) y = (xy ) and (x) (y ) = xy .
For the rst statement:
(x) y = (1) x) y
= (1)(x y )
= (x y )
Theorem 11.1(c)
M3
Theorem 11.1(c)
For the second statement notice rst that if
Math 2510: Real Analysis I Homework Solutions 10
Fall 2008
10 Natural Numbers and Induction
10.3 Prove that 12 + 22 + + n2 = 1 n(n + 1)(2n + 1) for all n N.
6
P (n) is the statement: 12 + 22 + + n2 = 1 n(n + 1)(2n + 1).
6
P (1) is the statement: 12 =
1
6
Math 2510: Real Analysis I Homework Solutions 8
Fall 2008
8 Cardinality
8.1 (a) Two sets S and T are equinumerous if there exists a bijection f : S T .
TRUE
(b) If a set S is nite, then S is equinumerous with In for some n N.
TRUE if 0 is in N; FALSE if 0
Math 2510: Real Analysis I Homework Solutions 7
Fall 2008
7 Functions
7.3 Find the range of each function f : R R.
(c) f (x) = x2 + 6x + 4
The graph is a parabola opening up. f (x) = 2x + 6 so the x-coordinate of the vertex is
x = 3. f (3) = 5. The range
Math 2510: Real Analysis I Homework Solutions 6
Fall 2008
6 Relations
6.6 In class.
6.8 Let A = cfw_a, b.
(a) How many elements are there in the set A A?
A A = cfw_(a, a), (a, b), (b, a), (b, b), so A A has 4 elements.
(b) How many possible relations are
Math 2510: Real Analysis I Homework Solutions 5
Fall 2008
5 Basic Set Operations
5.6 Let A and B be subsets of a universal set U . Simplify each of the following expressions.
(a) (A B ) (U \ A)
(A B ) (U \ A) = A B Ac = (A Ac ) B = U B = U
(b) (A B ) (U \
Math 2510: Real Analysis I Homework Solutions 4
Fall 2008
4 Techniques of Proof: II
4.4 Prove: There exists a rational number x such that x2 + 3x/2 = 1. Is this rational number unique?
Take x = 2. Then x is a rational number and x2 + 3x/2 = (2)2 + 3(2)/2
Math 2510: Real Analysis I Homework Solutions 3
Fall 2008
3 Techniques of Proof: I
3.3 Write the contrapositive of each implication.
(b) H is normal if H is not regular.
This is the same as: If H is not regular, then H is normal.
The contrapositive is: If
Math 2510: Real Analysis I Homework Solutions 2
Fall 2008
2 Quantiers
2.4 Write the negation of each statement.
(a) Some basketball players at Central High are short.
Negation: All players at Central High are tall.
(b) All of the lights are on.
Negation:
Math 2510 Test II
November 25, 2008
Show your work.
1. (2 points each)
(a) State the Continuum Hypothesis.
(b) What is Well-Ordering Property of the natural numbers.
(c) State the triangle inequality.
(d) If S R, what is the denition of inf S ?
(e) For x
Math 2510: Real Analysis I Homework Solutions 1
Fall 2008
1 Logical Connectives
1.4 Write the negation of each statement.
(b) The set of rational numbers is bounded.
Negation: The set of rational numbers is unbounded.
(c) The function f is injective and s
Math 2510: Real Analysis I Homework Solutions 13
Fall 2008
13 Topology of the Reals
13.1 Let S R Mark each statement True or False. Justify each answer.
(a) int S bd S =
TRUE: Suppose that y is in int S . Then there is an > 0 so that N (y, ) S . But then
Math 2510: Real Analysis I Homework Solutions 12
Fall 2008
12 The Completeness Axiom
12.3 For each subset of R, give its supremum and its maximum, if they exist. Otherwise, write none.
(a) cfw_1, 3
supcfw_1, 3 = maxcfw_1, 3 = 3
(c) [0, 4]
sup[0, 4] = max[
Math 2510: Real Analysis I Homework Solutions 7
Fall 2008
11 Ordered Fields
11.3b (-x) y = -(xy) and (-x) (-y) = xy. For the first statement: (-x) y = (-1) x) y = (-1)(x y) = -(x y) Theorem 11.1(c) M3 Theorem 11.1(c)
For the second statement notice first
Math 2510: Real Analysis I Homework Solutions 10
Fall 2008
10 Natural Numbers and Induction
10.3 Prove that 12 + 22 + + n2 = 1 n(n + 1)(2n + 1) for all n N.
6
P (n) is the statement: 12 + 22 + + n2 = 1 n(n + 1)(2n + 1).
6
P (1) is the statement: 12 =
1
6
Math 2510: Real Analysis I Homework Solutions 8
Fall 2008
8 Cardinality
8.1 (a) Two sets S and T are equinumerous if there exists a bijection f : S T . TRUE (b) If a set S is finite, then S is equinumerous with In for some n N. TRUE if 0 is in N; FALSE if
Math 2510: Real Analysis I Homework Solutions 7
Fall 2008
7 Functions
7.3 Find the range of each function f : R R. (c) f (x) = x2 + 6x + 4 The graph is a parabola opening up. f (x) = 2x + 6 so the x-coordinate of the vertex is x = -3. f (-3) = -5. The ran
Math 2510: Real Analysis I Homework Solutions 6
Fall 2008
6 Relations
6.6 In class.
6.8 Let A = cfw_a, b.
(a) How many elements are there in the set A A?
A A = cfw_(a, a), (a, b), (b, a), (b, b), so A A has 4 elements.
(b) How many possible relations are
Math 2510: Real Analysis I Homework Solutions 5
Fall 2008
5 Basic Set Operations
5.6 Let A and B be subsets of a universal set U . Simplify each of the following expressions.
(a) (A B ) (U \ A)
(A B ) (U \ A) = A B Ac = (A Ac ) B = U B = U
(b) (A B ) (U \
Math 2510: Real Analysis I Homework Solutions 4
Fall 2008
4 Techniques of Proof: II
4.4 Prove: There exists a rational number x such that x2 + 3x/2 = 1. Is this rational number unique? Take x = -2. Then x is a rational number and x2 + 3x/2 = (-2)2 + 3(-2)
Math 2510: Real Analysis I Homework Solutions 3
Fall 2008
3 Techniques of Proof: I
3.3 Write the contrapositive of each implication.
(b) H is normal if H is not regular.
This is the same as: If H is not regular, then H is normal.
The contrapositive is: If
Math 2510: Real Analysis I Homework Solutions 2
Fall 2008
2 Quantifiers
2.4 Write the negation of each statement. (a) Some basketball players at Central High are short. Negation: All players at Central High are tall. (b) All of the lights are on. Negation
Math 2510: Real Analysis I Homework Solutions 1
Fall 2008
1 Logical Connectives
1.4 Write the negation of each statement.
(b) The set of rational numbers is bounded.
Negation: The set of rational numbers is unbounded.
(c) The function f is injective and s