Description

CSCI 301 Homework 1 Solved

1. [2 points] Explicitly write out the contents of the following set:
�� �({1,2,3}): 2��
2. [3 points] Negate the following statement:
If x is a rational number and � ≠ 0, then tan(x) is not a rational number.
Prove each of the following statements.
Explicitly state which method of proof you are using: direct, contrapositive, contradiction
etc.
3. [5 points] Suppose � �ℤ. If a2 is not divisible by 4, then a is odd.
4. [5 points] Suppose, �, � � ℤ. If 4 | (�7 + �7), then a and b are not both odd.
5. [10 points] Given an integer a and b, then a2
(b+3) is even if and only if a is even or b is
odd.

CSCI 301 Homework 2 Solved

1) [4 points] Prove that 9 | (4#$ + 8) for every integer � ≥ 0.
2) [4 points] Prove that ∑ (8� − 5) $
/01 = 4�3 − � for every positive integer n.
3) [4 points] Prove that 13 + 23 + 33 + 43 + ⋯ + �3 = $($91)(3$91)
: for every positive
integer n.

CSCI 301 Homework 3 Solved

Use the method of proof by contradiction.
1. [4 points] Prove that √6 is irrational.
2. [4 points] If �, � �ℤ , then �) − 4� − 2 ≠ 0.

3. [4 points] Suppose Suppose A≠ Ø. Since Ø ⊆A×A, the set R= Ø is a relation on A. Is R
reflexive? Symmetric? Transitive? If a property does not hold, say why.
4. [4 points] Define a relation R on Z as xRy if and only if 4 | (x + 3y)
Prove R is an equivalence relation.
Describe its equivalence classes.