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Family Feud Review Math 5911: Discrete Mathematics for Educators July 11, 2011 Moore Family Feud

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Family Feud Review

Math 5911: Discrete Mathematics for Educators

July 11, 2011

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Question 1

What is an equivalence relation?

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Answer 1

AnswerAn equivalence relation is a relation that is reflexive, symmetric,and transitive.

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Question 2

Determine if the relation R on the set A = {a, b, c, d , e} is anequivalence relation where

R = {(a, a), (a, b), (a, c), (b, a), (b, b), (b, c), (c , a),

(c , b), (c, c), (c , d), (d , c), (d , d), (e, e)}.

Justify your answer.

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Answer 2

AnswerThe relation R is not an equivalence relation since R is nottransitive. We know R is not transitive since (b, c) ∈ R and(c , d) ∈ R, but (b, d) 6∈ R.

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Question 3

Find the first five terms of the sequence an = 4 + n2.

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Answer 3

Answer

a1 = 5a2 = 8a3 = 13a4 = 20a5 = 29

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Question 4

Find a formula for the nth term of the sequence

5, 6, 8, 11, 15, 20, · · · .

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Answer 4

Answer

a1 = 5an = an−1 + n − 1

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Question 5

What is an onto function?

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Answer 5

AnswerA function f : A −→ B is onto if for every y ∈ B there exists anx ∈ A such that f (x) = y .

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Question 6

Give an example of two sets A and B along with a functionf : A −→ B such that f is onto, but not one-to-one. Justify youranswer.

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Answer 6

AnswerLet A = {1, 2}, B = {z}, and f = {(1, z), (2, z)}. Notice that f isa function since every input in A has exactly one output in B. Thefunction is also onto because every element in B has acorresponding element in A that maps to it. However the functionis not one-to-one since f (1) = z and f (2) = z (which meansf (1) = f (2)), but 1 6= 2.

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Question 7

Consider the sets A = {1, 2, 3} and B = {a, b}. Determine |A×B|.

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Answer 7

AnswerRecall that |A× B| = |A||B| = 3(2) = 6. If you tried calculatingthe sets you would find six ordered pairs listed as follows

A× B = {(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)}.

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Question 8

Create a relation on the set A = {1, 2, 3, 4} that is symmetric andreflexive, but not transitive. Provide an explanation as to why yourexample satisfies those given properties.

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Answer 8

AnswerConsider the relationR = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1), (2, 4), (4, 2)}. Noticethat R is reflexive since for every a ∈ A we find (a, a) ∈ R.Similarly, R is symmetric since whenever (a, b) ∈ R we find(b, a) ∈ R. However, R is not transitive because (1, 2) ∈ R and(2, 4) ∈ R, but (1, 4) 6∈ R.

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Question 9

Suppose a|15 and a|40. Do we have enough information to know ifa|100? Justify your answer.

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Answer 9

AnswerYes, a|100 since a|4(15) and a|40 implies a|(4(15) + 40). The laststatement translates to a|100.

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Question 10

Using the Euclidean algorithm, determine the greatest commondivisor of 96 and 42.

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Answer 10

Answer

96 = 42(2) + 1242 = 12(3) + 612 = 6(2) + 0

This gives gcd(96, 42) = 6.

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Question 11

Let A = {∅, {a, b, c , d , e, f }, a, c , {d , e, f }, 17, {∅}}. Find |A|.

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Answer 11

Answer|A| = 7

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Question 12

Suppose |A| = 10 and S is the set of all subsets of A containingexactly 9 elements. Determine |P(A)− S |.

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Answer 12

AnswerNotice that |P(A)| = 210 and |S | = 10. Then|P(A)− S | = |P(A)| − |S | = 210 − 10 = 1014.

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Question 13

Assume gcd(a, b) = 8. Does there exist integers r and s such thatar + bs = 70? Give an explanation providing evidence for yoursolution.

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Answer 13

AnswerNotice that gcd(a, b) - 70. As a result, we know that there doesnot exist integers r and s such that ar + bs = 70.

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Question 14

Recall from a previous question that gcd(96, 42) = 6. Does thereexist integers c and d such that 96c + 42d = 18? If such integersdo not exist, give reasons supporting your answer. Otherwise,provide values of c and d that satisfy the given equation.

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Answer 14

Answer

42− 12(3) = 642− (96− 42(2))(3) = 6

42− 96(3) + 42(2)(3) = 642(7)− 96(3) = 6

42(7) + 96(−3) = 696(−3) + 42(7) = 6

3(96(−3) + 42(7)) = 3(6)96(−9) + 42(21) = 18

As a result, we find there is a solution with c = −9 and d = 21.

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Question 15

Consider the relation “is taller than” on the set of all people.Determine whether or not the relation is reflexive, symmetric,and/or transitive.

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Answer 15

AnswerThe relation is transitive, but it is neither reflexive nor symmetric.

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Question 16

Consider the relation R on Z given by (a, b) ∈ R if a ≡ b mod 3.Determine if R is an equivalence relation.

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Answer 16

AnswerNotice that R is reflexive since for every a ∈ Z we finda ≡ a mod 3, implying (a, a) ∈ R. For symmetric, let us begin byassuming (a, b) ∈ R. This gives a ≡ b mod 3, which is equivalentto b ≡ a mod 3. The last statement means (b, a) ∈ R, which tellsus R is symmetric. For transitive, we assume (a, b) ∈ R and(b, c) ∈ R. This implies a ≡ b mod 3 and b ≡ c mod 3. As aresult we find a ≡ c mod 3, which gives (a, c) ∈ R. Hence R istransitive. Since R is reflexive, symmetric, and transitive, we knowR is an equivalence relation.

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Question 17

Give an example of three consecutive composite integers.

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Answer 17

Answer14,15,16

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Question 18

Give an example of 17 consecutive composite integers.

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Answer 18

AnswerNotice 18! + 2, 18! + 3, 18! + 4, · · · , 18! + 17, 18! + 18 is a stringof 17 consecutive integers with the consecutive terms divisible by2, 3, 4, · · · , 17, 18. Hence we have our string of 17 consecutivecomposite integers.

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Question 19

Let U = {a, b, c , d , e, f , g , h, i} be a universal set with subsetsA = {b, d , e, f , g}, B = {d , f , h}, and C = {a, b, f , h, i}. Find B.

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Answer 19

AnswerB = {a, b, c , e, g , i}

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Question 20

Let U = {a, b, c , d , e, f , g , h, i} be a universal set with subsetsA = {b, d , e, f , g}, B = {d , f , h}, and C = {a, b, f , h, i}. Find(B ∪ C ) ∩ A.

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Answer 20

Answer(B ∪ C ) ∩ A = {a, b, d , f , h, i} ∩ {b, d , e, f , g} ={c , e, g} ∩ {b, d , e, f , g} = {e, g}

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Question 21

Give an example of a number x such that 15234234 ≡ x mod 100.

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Answer 21

Answerx = 34

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Question 22

Determine if the following string of digits is a valid example of anISBN number

1− 023− 25344− 9.

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Answer 22

Answer1(1) + 2(0) + 3(2) + 4(3) + 5(2) + 6(5) + 7(3) + 8(4) + 9(4) ≡1 + 0 + 6 + 1 + (−1) + (−3) + (−1) + (−1) + 3 ≡ 5 6≡ 9 mod 11This is not a valid ISBN number.

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