Home Work Assignmet

Question

1. How many different positive integers can be made from the digits {2, 4, 6, 8} if repetitions are allowed?

 

2. What is the telescoping form of f(x) = x⁴ + 7x³ - x² + 2x + 1 ?

 

3. Let r = {(a,a), (b,b), (c,c), (d,c)} be a relation on A= {a,b,c,d}. Which one of the following properties does r possess?

    a. Transitivity

    b. Injectivity

    c. Bijectivity

    d. Symmetry

    e. Surjectivity.

 

4. How many elements does the set P ({3,4,5,6} ∩ {1,3,5,7}) contain?

 

5. Let S(k) = 6S(k-1) + 25 and S(0) = -4. What is the solution for S(k) for all k ∈ ZZ ?

 

6. Let A = {2ⁿ | n ∈ {0,1,2,3}} and B = {3n| n ∈ {0,1,2,3}} . What is the set A ∪ B ?

 

7. If U = {2, 3, 5, 6, 8, 9, 11, 12} then which one of the following is the truth set of the proposition 2|x ∧ 3|x  over U ?

 

8. Let A = {1, 2, 3, 4} and L = [P (A); ∨,∧] by a lattice under the subset relation ⊆. What is the meet of A_1 ∧ A₂ of A₁ = {1, 2, 3} and A₂ = {1, 3, 4} ?

 

9. If Ζ→R then which one of the following is the domain of f ?

     a. R

     b. Z

     c. ∅

     d. N

     e. Q

 

10. Let p(x): x2 + x - 2 + 0, q(x) : x + 2 = 0, and r(x) : x > 0. What is the quantified propositions over Z is true?

Answer 1

 

Kenneth has given you three good answers.  I'll follow his lead and give you three more.

 

3. r = {(a,a), (b,b), (c,c), (d,c)}

Transitivity: This requires that if element x is related to element y and element y is related to element z, then element x must be related to element z.  This is trivial to observe if we start with x = a, b, or c.  Similarly, if x = d, y = c, then z = c is the only relation.  So transitivity holds.

 

Injectivity: This requires conceptually that we can't have two (or more) elements that are mapped to the same thing.  Conceptually, you can think of this as requiring that the inverse be a function.  However, both c and d are mapped to c.  If we were to consider the inverse of this relation, we wouldn't know what to map 'c' to, so it isn't injective.

 

Bijectivity: Means that it is both injective and surjective.  We've already demonstrated it isn't injective.

 

Symmetry: This requires that x relates to y if and only if y relates to x.  However, we see that d relates to c, but c does not relate to d.  Therefore, it isn't symmetric.

 

Surjectivity: This requires that the image and codomain are equal, or conceptually, that every element of the codomain is mapped to by at least one element.  However, nothing maps to 'd'.  So the function isn't surjective.

 

5. So this is a fun one, which is why I chose it.  First step is making an educated guess about the form of the general term.  We see that we're multiplying by 6 each time, so we may guess:

 

S(k) = a*6^k + b

 

So, we have two possible equations to help us find a and b.

 

S(k) = 6S(k-1) + 25

a*6^k + b = 6(a*6^k-1 + b) + 25

a*6^k + b = a*6^k + 6b + 25

-5b = 25

b = -5

 

Looking at our initial condition, we have

S(0) = -4

a*6⁰ + (-5) = -4

a - 5 = -4

a = 1

 

Therefore,

 

S(k) = 6^k - 5

 

Based on how we derived it, we can quickly see that it will meet both conditions.

 

7. U = {2, 3, 5, 6, 8, 9, 11, 12}.  What is the truth set of the proposition 2|x ∧ 3|x over U ?

So, we're looking for all the elements of U that are divisible by both 2 and 3, that is divisible by 6.  That would be {6,12}.

Answer 2

1. How many different positive integers can be made from the digits {2, 4, 6, 8} if repetitions are allowed?

INFINITELY MANY since there appears to be no limitation on size. e.g. 2, 22, 222, 2222, ... just for starters.

 

4. How many elements does the set P ({3,4,5,6} ∩ {1,3,5,7}) contain?  TWO (just 3 and 5 are in the intersection).

 

6. Let A = {2n | n ∈ {0,1,2,3}} and B = {3n| n ∈ {0,1,2,3}} . What is the set A ∪ B ?

A = {0,2,4,6}   B = {0,3,6,9}  A ∪ B includes everything that is in either (or both).  You can list them!

 

Perhaps other tutors will continue your work.