Question

Question Two
Let A = {1, 2, 3, 4} and B = {5, 7, 9}. Determine,
(i) A × B
(ii) B × A
(iii) Is A × B = B × A?
(iv) Is n (A × B) = n (B × A)?
(8 Marks)
Question Three
Determine whether each of the following form is a tautology Or a contradiction Or neither:
i) PQPQ
ii) PQPQ
iii) PQPQ
(9 Marks)
Question Four
Using principle of mathematical induction prove that
(1 + 2 + 3 + ... + n = n (n + 1) / 2 ( 3 Marks)

Answer 1

Step-by-step explanation:

Given sets are A = {1, 2, 3, 4} and

B = {5, 7, 9}

1)A×B={1, 2, 3, 4} ×{5, 7, 9}

=>A×B={(1,5),(1,7),(1,9),(2,5),(2,7),(2,9),. (3,5),(3,7),(3,9),(4,5),(4,7),(4,9)}--------(1)

n(A×B)=12-------(2)

B×A={5, 7, 9}×{1, 2, 3, 4}

=>{(5,1),(5,2),(5,3),(5,4),(7,1),(7,2),(7,3),(7,4),(9,1),(9,2),(9,3),(9,4)}--------(3)

n(B×A)=12------(4)

From (1)&(3)

A×B≠B×A

from(2)&(4)

n(A×B)=n(B×A)

2)Given that

1+2+3+---+n=n(n+1)/2

Put n=1 then

LHS=1

RHS=1(1+1)/2=2/2=1

LHS=RHS is true for n=1

now put n=k then 1+2+3+---+K=K(K+1)/2

LHS =RHS is true for n=k

now put n=k+1 then

LHS=1+2+3----(k+1)

RHS=(k+1)(k+1+1)/2

=>(k+1)(k+2)/2

LHS =RHS is true for n=k+1

1+2+3+---+n=n(n+1)/2

Answer 2

Answer:

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