VLGA Theorems term 1

Question 1

De morgans Law

Answer 1

(i) (AUB)’ = A’∩B’

(ii) (A∩B)’ = A’UB’

Question 2

Associative Law

Answer 2

For any three sets

(i) A∩(B∩C) = (A∩B)∩C = A∩B∩C
(ii) AU(BUC) = (AUB)UC = AUBUC

Question 3

corollary of de morgans law

Answer 3

(AUBUC)’ = A’∩B’∩C’

Question 4

Distributive Law

Answer 4

For any three sets A, B and C:

i) A∩(BUC) = (A∩B)U(A∩C
(ii) AU(B∩C) = (AUB)∩(AUC)

Question 5

Principle of mathematical induction (1)

Answer 5

Suppose p(n) is a statement involving n∈ℕ  and that:
(i) p(1) is true
AND
(ii) For each k∈ℕ, we have p(k) is true => p(k+1) is true
Then p(n) is true for all n∈ℕ

Question 6

Principle of mathematical induction (2)

Answer 6

Suppose p(n) is a statement involving n∈ℕ with n≥a suppose that:
(i) p(a) is true
AND
(ii) p(k) is true => p(k+1) is true for all k≥a
Then p(n) is true for all n≥a

Question 7

number of inverses theorem

Answer 7

Any nxn matrix has most one inverse

Question 8

Elementary Row operation Theorem

Answer 8

EROs do not alter the solution of a system of linear equations

Question 9

Expansion of inverse matrices Theorem

Answer 9

Suppose that A and B are invertible nxn matrices, then the matrix AB is also invertible and (AB)⁻¹ = A⁻¹B⁻¹

Question 10

inverse of a 2x2 matrix

Answer 10

Let A (a b 
           c d)
be a general 2x2 matrix. Then,
(i) A is invertible iff (ad-bc)!=0
and in this case
(ii) A⁻¹ = - 1/(ad-bc)(d -b
                              -c a)

Question 11

Inverse of a general nxn matrix

Answer 11

To find the inverse of a general nxn matrix A:
(i) write down the augmented nxn matrix (A|Iₙ)
2. perform EROs with the aim of obtaining Iₙ in the LHS
(A|Iₙ) -> (Iₙ|B)
either this fails because atleast one row of zeros hence A is not invertible
Or its possible in which case B = A⁻¹

Question 12

Vector Product

Answer 12

Vector product in cartesian form:

Let u = GAVE UP BECAUSE NO POINT