family feud review - armstrong, a georgia university · 2011-07-11 · family feud review math...
TRANSCRIPT
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 4
Find a formula for the nth term of the sequence
5, 6, 8, 11, 15, 20, · · · .
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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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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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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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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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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 22
Determine if the following string of digits is a valid example of anISBN number
1− 023− 25344− 9.
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