Search & Filtering
In general, a statement involving n variables x1 , x 2 ,…, x n can be denoted by P( x1 , x 2 ,…, x n ) and is the value of the predicate P at the n-tuple (x1 , x2 ,…, x n ) . (a) Suppose Q(x , y) denotes the statement “ x=y+3 .” What are the truth values of the propositions Q(1,2) and Q(3,0)? (b) Let R(x , y ,
{1,2,3,4,6,9}find maximal and minimal and least and greatest number
5.
Which relation on the set {1, 2, 3, 4} is an equivalence relation and contain {(1, 2), (2, 3), (2, 4), (3, 1)}.
6.
Find the transitive closures of the relation {(1, 1), (1,4), (2,1), (2,3), (3,1), (3, 2), (3,4), (4, 2)} on the set {1, 2, 3, 4}.
7.
Let R be the relation on the set {1, 2, 3, 4, 5} containing the ordered pairs (1, 1), (1, 2), (1, 3), (2, 3), (2, 4), (3, 1), (3, 4), (3, 5), (4, 2), (4, 5), (5, 1), (5, 2), and (5, 4).
Find a) R3 b) R4
8.
Which of these relations on the set of all people are equivalence relations? Determine the properties of an equivalence relation that the others lack.
a) {(a, b) | a and b are the same age}
b) {(a, b) | a and b have the same parents}
c) {(a, b) | a and b speak a common language}
2.
Find the transitive closures of these relations on {1, 2, 3, 4}.
a) {(1, 2), (2,1), (2,3), (3,4), (4,1)}
b) {(2, 1), (2,3), (3,1), (3,4), (4,1), (4, 3)}
c) {(1, 2), (1,3), (1,4), (2,3), (2,4), (3, 4)}
d) {(1, 1), (1,4), (2,1), (2,3), (3,1), (3, 2), (3,4), (4, 2)}
3.
Find the smallest relation containing the relation {(1, 2), (1, 4), (3, 3), (4, 1)} that is
a) reflexive and transitive.
b) symmetric and transitive.
c) reflexive, symmetric, and transitive.
4.
Which of these relations on {0, 1, 2, 3} are equivalence relations? Determine the properties of an equivalence relation that the others lack.
a) {(0, 0), (1, 1), (2, 2), (3, 3)}
b) {(0, 0), (0, 2), (2, 0), (2, 2), (2, 3), (3, 2), (3, 3)}
c) {(0, 0), (1, 1), (1, 2), (2, 1), (2, 2), (3, 3)}
d) {(0, 0), (1, 1), (1, 3), (2, 2), (2, 3), (3, 1), (3, 2),(3, 3)}
e) {(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0),(2, 2), (3, 3)}
1.Let R be the relation on the set {0, 1, 2, 3} containing the ordered pairs (0, 1), (1, 1), (1, 2), (2, 0), (2, 2), and (3, 0). Find the
a) reflexive closure of R.
b) symmetric closure of R.
Construct the Combinatorial Circuit of the given output.
- (p∨q) 2. (p∧ ~q)
- (p→~q)→(p∨q)
Construct the truth table of the following proposition.
1. (p∧q)→(p∨q)
2. ~(p→q)
3. (p∨q)∧~p
Consider the following list of numbers:
1 7 8 14 20 42 55 67 78 101 112 122 170 179 190
Apply binary algorithms in order to find the target which are number 42 and number 82. Show your step and calculate the total number of comparisons for each item.
Prove that the set G={0,1,2,3,4,5} is an abelian group with respect to the
multiplication modulo 6.
Let X = {1, 2, 3, 4, 6, 8, 12, 24} and R be a division relation defined on X. Find
the Hasse diagram of the poset <X, R>.
