Search & Filtering
Let R be a relation on the set { a, b, c, d }
R = { (a, b), (a, c), (a, d), (c, b), (c, d), (d, b)}.
Identify the properties satisfied on this given relation.
Consider a relation R on a set A = { 2, 4, 7 }.
Given the relation R = { (2, 2), (2, 4), (2, 7), (4, 7}. Find:
1. Complement of a Relation
_____________________________________________________
_____________________________________________________
2. Inverse of a Relation
__________________________________________________________________________________________________________
3. Composite Product R o S and S o R ; S = { (1, 2), (2, 4), (2, 7) }
1. Let S be the set of all strings of English letters. Determine whether these relations are reflexive, irreflexive, symmetric, antisymmetric, and/or transitive.
a) R1 = {(a, b) | a and b have no letters in common}
b) R2 = {(a, b) | a and b are not the same length}
c) R3 = {(a, b) | a is longer than b}
Let a_{k} = 3^{k} + k - 2 for all k \geq 0.
Write down the values of a_{1}, a_{2} and a_{3}.
Write down the values of A(1), A(2) and A(3) defined by the recurrence relation: A(0) = -1, A(k) = 3A(k - 1) - 2k + 7, k \geq 1
Show that A(k) = a_{k} is a solution of the recurrence relation for all values of k \geq 1.
Write down all derangements of the set \left\{ a,b,c,d \right\} and show that the number of derangements is the same as predicted by the recurrence D(n) = (n - 1)(D(n - 2) + D(n - 1)) with initial values D(1) = 0 and D(2) = 1. Hint: a derangement is a permutation of an ordered set where no element is in the same place as before. Example: \left\{ b,a,d,c \right\} is a derangement of \left\{ a,b,c,d \right\} because all of the letters positions have changed.
Consider a relation R on a set A = { 2, 4, 7 }.
Given the relation R = { (2, 2), (2, 4), (2, 7), (4, 7}. Find:
1. Complement of a Relation
_____________________________________________________
_____________________________________________________
2. Inverse of a Relation
__________________________________________________________________________________________________________
3. Composite Product R o S and S o R ; S = { (1, 2), (2, 4), (2, 7) }
If 1 + 2 = 3, then 3 is odd.
There is a set X = {1, 2, 3, 4, 5}. Find the sets A, B, C such that A ⊆ X, B ⊆ X,
C ⊆ X,
(A ∪ B) ⊆ (A ∩ C)
and C 6⊆ B.
1. Let A = { 2, 3 }, and B = { 3, 4, 5, 6 }
Let R be a relation from A to B where
R = {(x, y) “x exactly divides y”}
Enumerate the following:
A. elements of R; R = { ______, ______, ______, ______ }
rewrite R as a table.
A B
2. Let R be a relation on A = { 1, 2, 3 } defined by
R = { (x, y) l x ≤ y ˄ x, y ∈ A }
Identify the domain and range of the relation R.
Solve the recurrence relation A(n) = 6A(n - 1) - 11A(n - 2) + 6A(n - 3) subject to initial values A(1) = 2, A(2) = 6, A(3) = 20.
