MAT229: Homework on Functions (part 1)
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Homework on Functions (part 1)
Problem 1. Prove that A B A B for all sets A and B.
Problem 2. Prove that A B if and only if A B = .
Problem 3. Let R be the set of all real numbers and f : R R be dened f (x) = x2 .
Partial Orders
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Partial Orders
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Partial Orders
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A relation R on a se
Methods of Proof
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Methods of Proof
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Methods of Proof
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Theorem
A statement that can b
Set Operations
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Set Operations
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Set Operations
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Let A and B be sets.
The union of A
Relations
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Relations
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Relations
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Let A and B be sets. A binary relation from A
MAT229: Homework on Logic
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Homework on Logic
Problem 1. Prove that and are commutative; that is p q q p and similarly for
.
Problem 2. Prove that distributes over . That is, show that p(qr) (pq)(pr).
Problem 3. Prove that p (p) F and p (p) T .
Problem 4.
Sets
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Sets
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Sets
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A set is a collection of objects. The objects are called elements or
m
Predicates and Quantifiers
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Predicates and Quantifiers
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Predicates and Quantifiers
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MAT229: Homework on Functions (part 2)
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Homework on Functions (part 2)
Problem 1.
Suppose that X = cfw_p, q, r, Y = cfw_a, b, c, d and Z = cfw_1, 2, 3, 4. Let
g : X Y be dened by the set of ordered pairs cfw_(p, a), (q, b), (r, c) and f : Y Z be
dened by
MAT229: Homework on Predicates and Quantiers
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Homework on Predicates and Quantiers
Problem 1. Let Q(x, y) denote the statement x is the capital of y. What are the truth
values of the following?
a. Q(Denver, Colorado)
b. Q(Detroit, Michigan)
c. Q(Massachu
MAT229: Homework on Sets
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Homework on Sets
Problem 1. Prove that distributes over .
Problem 2. Prove that if A B then B A .
Problem 3.
answer.
What can you conclude about A and C if A B and B C? Prove your
Problem 4. Dene the universal set U = the set of
Logic
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Logic
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Logic
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Propositions
A proposition is a statement that is either true or
Functions
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Functions
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Functions
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Let A and B be sets. A function f from A to B