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The relation P does not satisfy trichotomy. Which ordered pairs should be included in P so that an extended relation P1 (say) would satisfy trichotomy? (For trichotomy, each element of B must be paired with each other different element in B to form elements of P1. For example, we see that b {a} but neither (b, {a}) nor ({a}, b) are elements of P, but at least one of these elements should be an element of P1. We include (b, {a}) in P1: P1 = {(a, b), (b, {a, b}), ({a, b}, a), ({b}, a), (a, {a}), (b, {a}), ...}.
Choose the alternative that provides all the missing ordered pairs that should be included as elements of P1 in order for P1 to be a relation that satisfies trichotomy. 1. (b, {a}), (b, {b}), (b, a), ({a}, {a, b}) & ({a, b}, {b})
2. ({a}, b), (b, {b}), ({b}, {a, b}) & ({a, b}, {a})
3. (b, {a}), ({b}, b), ({a}, {b}), ({b}, {a, b}) & ({a, b}, {a})
4. (b, {a}), (b, {b}), ({a}, {b}) & ({a, b}, {a, b})
Which one of the following relations represents the composition relation P ○ P (ie P; P)? 1. {(a, {a, b}), (b, a), ({a, b}, a), ({b}, {a}), (a, {a})} 2. {(a, {a, b}), (b, a), ({a, b}, a), ({b}, {a})} 3. {(a, {a, b}), (b, a), ({a, b}, {a}), ({a, b}, b), ({b}, b), ({b}, {a})} 4. {(a, {a, b}), (b, a), ({a, b}, {a}), ({a, b}, b), ({b}, {a})}
Consider the following relation on set B = {a, b, {a}, {b}, {a, b}}: P = {(a, b), (b, {a, b}), ({a, b}, a), ({b}, a), (a, {a})}.
Which one of the following alternatives represents the range of P (ran(P))? 1. {a, b, {a}, {a, b}}
2. {a, b, {a}, {b}, {a, b}}
3. {a, b, {b}, {a, b}}
4. {a, b, {a, b}}
Let T be a relation from A = {0, 1, 2, 3} to B = {0, 1, 2, 3, 4} such that (a, b) T iff b2 – a2 is an odd number. (A, B U = Z.) (Hint: Write down all the elements of T. For example, if 4 B and 1 A then 42 – 12 = 16 – 1 = 15 which is an odd number, thus (1, 4) T.) Answer questions 5 and 6 by using the defined relation T. Question 5 Which one of the following alternatives provides only elements belonging to T? 1. (3, 1), (4, 1), (3, 2) 2. (0, 1), (2, 4), (2, 3) 3. (3, 0), (1, 2), (3, 4) 4. (1, 0), (1, 2), (1, 3) Question 6 Which one of the following statements regarding the relation T is true? 1. T is transitive. 2. T is symmetric. 3. T is antisymmetric. 4. T is irreflexive.
Fourty (40) students go to a party wearing red, white and blue. Of these students, 17 wear red, 22 wear white, 25 wear blue. (Students do not necessarily wear only one colour.) Furthermore, 7 wear red and white, 12 wear blue and white, and 9 wear red and blue. Which one of the following alternatives is true? (Hint: first calculate the value of x, the unknown) 1. 5 students wear red only. 8 students wear white and blue, but not red. 3 students wear red and white, but not blue. 2. 2 students wear red only. 11 students wear white and blue, but not red. 6 students wear red and white, but not blue. 3. 2 students wear red only. 8 students wear white and blue, but not red. 3 students wear red and white, but not blue. 4. 5 students wear red only. 11 students wear white and blue, but not red. 6 students wear red and white, but not blue.
The sum of the values of the degree 𝜎(V) taken over all the vertices v of a graph .G= (V, E), is equal to twice the number of edges i.e.
∑ 𝜎(𝑣) = 2|E| v∈V.
Write a proof to validate the equation.
What are the two types of indirect proofs? Explain through an example for each type.
Suppose that p and q are any statements. By constructing the truth tables, show that the statement ¬ (p V q) & (¬ p) ∧ (¬ q) are logically equivalent
Determine whether ( 𝑝∨𝑞)∧(𝑝→𝑟)∧( 𝑞→𝑠)→𝑟∨𝑠 is a Tautology or a contradiction
a) Suppose p, q and r mean “Kelly is at home, Hannah is at home and Sunny is at home then interpret the meaning of the statement p ∧∼q and (p ∧ q) ∧ r
(b) You should pay the fares for tickets only if you plan to visit northern Pakistan in this winter. (Convert the sentence in symbolic notation)
(c) If the lift in our office install then one can use the lift (write a converse of the statement in symbolic notation as well as a statement)
(d) If we do not make a plan to plant marigold flowers then it is not suitable season for them. (Write the contrapositive statement in symbolic as well as a statement)
(e) If it is right angled triangle then Pythagoras rule applies. (Convert the statements in biconditional statement in symbolic as well as a statement)
