Discrete Math for Comp Sci Unit 4
If 𝑅 is a relation from a set 𝑋 to a set 𝑌, we label the [ 1 ] of a matrix with the elements of 𝑋, and the [ 2 ] with the elements of 𝑌.
1. rows 2. columns
The matrix of a relation 𝑅 is 𝐴= [ 1 0 0 1 0 ] 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 Determine if the relation is
yes to: reflexive symmetric transitive equivalence
With regard to the matrix 𝑀 of a relation 𝑅, to determine if 𝑅 is antisymmetric we need to check if
∀𝑖∀𝑗(𝑚𝑖,𝑗≠0→𝑚𝑗,𝑖=0)
A relation is reflexive if
∀𝑥(𝑥𝑅𝑥)
A relation is antisymmetric if
∀𝑥∀𝑦(𝑥𝑅𝑦∧𝑦𝑅𝑥→𝑥=𝑦)
A relation is transitivie if
∀𝑥∀𝑦∀𝑧(𝑥𝑅𝑦∧𝑦𝑅𝑧→𝑥𝑅𝑧)
In the digraph of a relation, a loop occurs when
∃𝑥(𝑥,𝑥)∈𝑅
A positive integer 𝑛 is prime if and only if 𝑛 does not have proper divisors less than or equal to
√𝑛 (square root)
gcd(𝑎,𝑏)⋅lcm(𝑎,𝑏) is equal to
𝑚𝑛
Find the matrix of the relation on 𝑋={1,2,3} 𝑅={(1,2),(1,3),(2,1),(3,1),(3,3)} A=
0 1 1 1 0 0 1 0 1
In the binary number 10110(1), the underlined 1 is in the
1's place
Find lcm( 4158 , 15288 )
1,513,512
A prime number is a number greater than [ 1 ] whose only positive divisors are [ 2 ]
1. 1 2. 1 and itself
Determine the equivalence classes of the following equivalence relation on 𝑋={1,2,3,4,5,6} 𝑅={(𝑥,𝑦)∣𝑥+𝑦 is even} Note: write the elements in each class in increasing order, separated by a comma and no space. E.g. 1,3,5 [1]={ } [2]={ } [3]={ } [4]={ } [5]={ } [6]={ }
1. 1,3,5 2. 2,4,6 3. 1,3,5 4. 2,4,6 5. 1,3,5 6. 2,4,6
Determine the equivalence classes of the following equivalence relation on 𝑋={1,2,3,4,5} 𝑅={(𝑥,𝑦)∣3 divides 2𝑥+𝑦} Note: write the elements in each class in increasing order, separated by a comma and no space. E.g. 1,3,5 [1]={ } [2]={ } [3]={ } [4]={ } [5]={ }
1. 1,4 2. 2,5 3. 3 4. 1,4 5. 2,5
Determine the equivalence classes of the following equivalence relation on 𝑋={1,2,3,4,5,6} 𝑅={(𝑥,𝑦)∣4 divides 3𝑥+𝑦} Note: write the elements in each class in increasing order, separated by a comma and no space. E.g. 1,3,5 [1]={ } [2]={ } [3]={ } [4]={ } [5]={ } [6]={ }
1. 1,5 2. 2,6 3. 3 4. 4 5. 1,5 6. 2,6
Determine the equivalence classes of the following equivalence relation on 𝑋={1,2,3,4,5,6} 𝑅={(𝑥,𝑦)∣5 divides 2𝑥+3𝑦} Note: write the elements in each class in increasing order, separated by a comma and no space. E.g. 1,3,5 [1]={ } [2]={ } [3]={ } [4]={ } [5]={ } [6]={ }
1. 1,6 2. 2 3. 3 4. 4 5. 5 6. 1,6
Let m, n, and d be integers. Show that, if d|m and d|n, then d|m-n.
1. d|m / {given} 2. d|n / {given} 3. {m=dq for some q} / by {1} and definition of d|m 4. {n=dr for some r} / by {2} definition of d|n 5. m-n=d(q-r) / by {3} and {4} 6. d|m-n / by {5} and {definition of d|m-n}
Let m, n, h, and k be integers. Show that, if h|m and k|n, then hk|mn.
1. h|m / {given} 2. k|n / {given} 3. {m=hq for some q} / by {1} and definition of h|m 4. {n=kr for some r} / by {2} definition of k|n 5. mn=hqkr / by {3} and {4} 6. mn=hk(qr) / by {5} and {commutativity of the product} 7. hk|mn / by {6} and {definition of hk|mn}
A partition of a set 𝑋 is a [ 1 ] S of [ 2 ] such that every element of X belongs to [ 3 ] member of S
1. set 2. non-empty subsets of X 3. exactly one
To emphasize that the number 101101 should be regarded as a binary number, we will write
101101 \/ 2
Find lcm( 8232 , 8820 )
123,480
Conver 2033\/10 in hexadecimal form. 2033 :16=
127 1 127 7 15 7 0 7 7F1
Find gcd( 7800 , 6084 )
156
Conver 2845\/10 in hexadecimal form. 2845 :16=
177 13 177 11 1 11 0 11 B1D
In the binary number 1010(1)0, the underlined 1 is in the
2's place
Convert 3478\/10 to hexadecimal form. 3478 :16=
217. 6 217. 13. 9 13. 0. 13 D96
Find lcm( 1650 , 2420 )
36,300
In the binary number 100(1)00, the underlined 1 is in the
4's place
Find gcd( 4158 , 15288 )
42
Conver 85\/10 in binary form. 85 : 2 =
42. 1 42. 21. 0 21. 10. 1 10. 5. 0 5. 2. 1 2. 1. 0 1. 0. 1 1010101
Conver 93\/10 in binary form. 93. : 2 =
46. 1 46. 23. 0 23. 11. 1 11. 5. 1 5. 2. 1 2. 1 0 1. 0. 1 1011101
Conver 103\/10 in binary form. 103 : 2 =
51. 1 51. 25. 1 25. 12. 1 12. 6. 0 6. 3. 0 3. 1. 1 1. 0. 1 1100111
Find gcd( 13860 , 12936 )
924
In the hexadecimal system, the number 10 is represented by
A
In the hexadecimal system, the number 12 is represented by
C
In the hexadecimal system, the number 13 is represented by
D
In the hexadecimal system, the number 14 is represented by
E
In the hexadecimal system, the number 15 is represented by
F
Given a relation 𝑅 on a set 𝑋, the matrix associated with 𝑅 is uniquely determined. (t or f)
False
If the prime factorization of 𝑚 and 𝑛 are respectively m=p1a1p2a2⋯pkak n=p1b1p2b2⋯pkbk then gcd(m,n) is equal to
P1min(𝑎1,𝑏1)P2min(𝑎2,𝑏2)⋯Pkmin(𝑎𝑘,𝑏𝑘)
With regard to the matrix 𝑀 of a relation 𝑅, to determine if 𝑅 is reflexive we need to check if
The diagonal of 𝑀 has all non-zero entries
With regard to the matrix 𝑀 of a relation 𝑅, to determine if 𝑅 is symmetric we need to check if
The matrix 𝑀 is symmetric
A relation can be given by simply specifiying which ordered pairs belong to the relation (t or f)
True
Two equivalence classes of an equivalence relation either coincide or are disjoint.
True
A relation 𝑅 from a set 𝑋 to a set 𝑌 is
a subset of 𝑋×𝑌
Let 𝑅 be the relation on the set of {1,2,3,4,5} described by the following digraph: (triangle connect in middle) 1O 2O O / 3 \ O4---------5O The 𝑅 is (check all that apply)
antisymmetric reflexive
A binary digit is also referred as a
bit
For a partial order 𝑅, if, for two elements 𝑥,𝑦, we have that 𝑥𝑅𝑦 or 𝑦𝑅𝑥, then 𝑥 and 𝑦 are said to be
comparable
A digraph consists of (check all that apply)
directed edges verticles loops
With regard to the matrix 𝑀 of a relation 𝑅, to determine if 𝑅 is transitive we need to check if (check all that apply)
for each zero off-diagonal entry in 𝑀 the corresponding entry in 𝑀2 is also zero for each non-zero off-diagonal entry in 𝑀2 the corresponding entry in 𝑀 is also non-zero
Determine if the given relation is an equivalence relation on 𝑋={1,2,3,4,5} 𝑅={(1,1),(2,2),(3,3),(4,4),(5,5),(1,3),(3,1),(3,4),(4,3)}
no
Determine if the given relation is an equivalence relation on 𝑋={1,2,3,4,5} 𝑅={(1,1),(2,2),(3,3),(4,4),(5,5),(1,5),(1,3),(3,5)}
no
Determine if the given relation is an equivalence relation on 𝑋={1,2,3,4,5} 𝑅={(1,1),(2,2),(3,3),(4,4)}
no
Determine if the given relation is an equivalence relation on 𝑋={1,2,3,4,5} 𝑅={(𝑥,𝑦)∣3 divides 𝑥+3𝑦}
no
Determine if the given relation is an equivalence relation on 𝑋={1,2,3,4,5} 𝑅={(𝑥,𝑦)∣4 divides 𝑥+𝑦}
no
Determine if the given relation is an equivalence relation on 𝑋={1,2,3,4,5} 𝑅={(𝑥,𝑦)∣𝑥 and 𝑦 are both odd}
no
Let 𝑅 be the relation on the set of positive integers defined by (𝑥,𝑦)∈𝑅 if 4 divides 𝑥+2𝑦. Then 𝑅 is (check all that apply)
none of the above (or below)
Let 𝑅 be the relation on the set of real numbers defined by (𝑥,𝑦)∈𝑅 if 𝑥^3⩽𝑦^3. Then 𝑅 is (check all that apply)
partial order antisymmetric reflexive transitive total order
Let 𝑅 be the relation on a power set P(X) defined by (𝐴,𝐵)∈𝑅 if 𝐴⊆𝐵. Then 𝑅 is (check all that apply)
reflexive antisymmetric transitive partial order
An equivalence relation is a relation that is (check all that apply)
reflexive symmetric transitive
Check the properties characterizing a partial order
reflexive transitive antisymmetric
Check the properties characterizing a total order
reflexive transitive antisymmetric all pairs x,y are comparable
Let 𝑅R be the relation on the set of {1,2,3,4,5,6} described by the following digraph: (triangle inside circle, 3 circles, 1 circle) 1O3 2 5 O 4O 6 The 𝑅 is (check all that apply)
reflexive transitive symmetric
Let 𝑅 be the relation on the set of positive integers defined by (𝑥,𝑦)∈𝑅 if 3 divides 𝑥+𝑦. Then 𝑅 is (check all that apply)
symmetric
Let 𝑅 be the relation on the set of positive integers defined by (𝑥,𝑦)∈𝑅 if 𝑥𝑦⩽2. Then 𝑅 is (check all that apply)
symmetric
Let 𝑅 be the relation on the set of {𝑎,𝑏,𝑐,𝑑} described by the following digraph: (Animated circles) aO cOb dO The 𝑅 is (check all that apply)
symmetric
Let 𝑅 be the relation on the set of real numbers defined by (𝑥,𝑦)∈𝑅 if 𝑥⩽𝑦−1. Then 𝑅 is (check all that apply)
transitive antisymmetric
Let 𝑅 be the relation on the set of real numbers defined by (𝑥,𝑦)∈𝑅 if 𝑥^2⩽𝑦^2. Then 𝑅 is (check all that apply)
transitive reflexive
Let 𝑅 be the relation on the set of positive integers defined by (𝑥,𝑦)∈𝑅 if 3 divides 𝑥+2𝑦. Then 𝑅 is (check all that apply)
transitive reflexive symmetric
Let 𝑅 be the relation on the set of {𝑎,𝑏,𝑐} described by the following digraph: (3 circles) aObOcO The 𝑅 is (check all that apply)
transitive reflexive symmetric partial order antisymmetric
Let 𝑅 be the relation on the set of positive integers defined by (𝑥,𝑦)∈𝑅 if 𝑥𝑦⩾1. Then 𝑅 is (check all that apply)
transitive symmetric reflexive
Determine if the given relation is an equivalence relation on 𝑋={1,2,3,4,5} 𝑅={(1,1),(2,2),(3,3),(4,4),(5,5),(1,3),(3,1)}
yes
Determine if the given relation is an equivalence relation on 𝑋={1,2,3,4,5} 𝑅={(𝑥,𝑦)∣4 divides 𝑥+3𝑦}
yes
Determine if the given relation is an equivalence relation on 𝑋={1,2,3,4,5} 𝑅={(𝑥,𝑦)∣𝑥 and 𝑦 are both even or both odd}
yes
The matrix of a relation 𝑅 is 𝐴=[ 1. 1. 1. 1. 1 ] 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 [ 0 1 1 1 0 ] Determine if the relation is
yes to: antisymmetric transitive
The matrix of a relation 𝑅 is 𝐴= [ 1 0 0 1 1 ] 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 0 0 0 1 1 Determine if the relation is
yes to: reflexive antisymmetric
The matrix of a relation 𝑅 is 𝐴= [ 1 0 0 1 1 ] 0 1 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 0 1 1 Determine if the relation is
yes to: reflexive antisymmetric transitive partial order
The matrix of a relation 𝑅 is 𝐴= [ 1 1 0 1 1 ] 0 1 0 0 0 1 1 1 1 1 0 1 0 1 1 0 0 0 0 1 Determine if the relation is
yes to: reflexive antisymmetric transitive partial order
The matrix of a relation 𝑅 is 𝐴= [ 1 1 1 1 1 ] 0 1 1 1 1 0 0 1 1 1 0 0 0 1 0 0 0 0 0 1 Determine if the relation is
yes to: reflexive antisymmetric transitive partial order
The matrix of a relation 𝑅 is 𝐴= [ 1 1 0 1 1 ] 0 1 0 0 0 1 1 1 1 1 0 1 0 1 1 0 1 0 0 1 Determine if the relation is
yes to: reflexive antisymmetric transitive partial order total order
The matrix of a relation 𝑅 is 𝐴=[ 1 0 0 1 0 ] 0 1 0 0 1 0 0 1 1 0 1 0 1 1 0 [ 0 1 0 0 1 ] Determine if the relation is
yes to: reflexive symmetric
The matrix of a relation 𝑅 is 𝐴= [ 1 0 0 0 1 ] 0 1 0 0 0 0 0 1 1 0 0 0 1 1 0 1 0 0 0 1 Determine if the relation is
yes to: reflexive symmetric transitive equivalence
The matrix of a relation 𝑅 is 𝐴= [ 1 0 0 1 1 ] 0 1 0 0 0 0 0 1 0 0 1 0 0 1 1 1 0 0 1 1 Determine if the relation is
yes to: reflexive symmetric transitive equivalence
