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Discrete Mathematics
Suppose that the number of bacteria in a colony triples every hour. Let Bn denote the number of bacteria in the colony after n hours.
(a) Set up a recurrence relation for Bn.
(b) If 100 bacteria are used to begin a new colony, how many bacteria will be in the colony in 10 hours?
(a) Set up a recurrence relation for Bn.
(b) If 100 bacteria are used to begin a new colony, how many bacteria will be in the colony in 10 hours?
Discrete Mathematics
1) A person deposits 1000 USD into an account that yields 9 percent interest compounded annually. Let An denote the amount of money in the account after n years.
(a) Set up a recurrence relation for An.
(b) Find an explicit formula for An.
(c) How much money will be in the account after 100 years?
(a) Set up a recurrence relation for An.
(b) Find an explicit formula for An.
(c) How much money will be in the account after 100 years?
Discrete Mathematics
Let f : A → B be a function and σ an equivalence relation on B. Define a relation ρ on A as: a ρ a' if and only if f(a) σ f(a').
1. Prove that ρ is an equivalence relation on A.
2. Define a map f : A/ρ → B/σ as [a]ρ 7→ [f(a)]σ. Prove that f is injective.
3. Prove or disprove: If f is a bijection, then so is f.
4. Prove or disprove: If f is a bijection, then so is f.
Discrete Mathematics
Construct a proof for the five color theorem for every planar graph.
2. Discuss how efficiently Graph Theory can be used in a route planning project for a vacation trip from Colombo to Trincomalee by considering most of the practical situations (such as mileage of the vehicle, etc.) as much as you can. Essentially consider the two fold,
- Routes with the shortest distance (Quick route travelling by own vehicle)
- Route with the lowest cost
2. Discuss how efficiently Graph Theory can be used in a route planning project for a vacation trip from Colombo to Trincomalee by considering most of the practical situations (such as mileage of the vehicle, etc.) as much as you can. Essentially consider the two fold,
- Routes with the shortest distance (Quick route travelling by own vehicle)
- Route with the lowest cost
Discrete Mathematics
Let U = {English, French, History, Math, Physics, Chemistry, Psychology, Drama},
A = {English, Chemistry, French, Psychology},
B = {Math, Physics, History, French, Psychology},
and
C = {Drama, Chemistry, History}.
Find the following.
n[(A ∪ B) ∪ C]
A = {English, Chemistry, French, Psychology},
B = {Math, Physics, History, French, Psychology},
and
C = {Drama, Chemistry, History}.
Find the following.
n[(A ∪ B) ∪ C]
Discrete Mathematics
There are 6 routes from Delhi to Mumbai and 12 routes from Mumbai to Bangalore. In how many ways can you travel from Delhi to Bangalore via Mumbai?
18
72
36
16
18
72
36
16
Discrete Mathematics
1. Let Q(x) denote the statement “x=x+1” What is the truth value of the
quantification ∃x Q(x), where the domain consists of all real
numbers?
quantification ∃x Q(x), where the domain consists of all real
numbers?
Discrete Mathematics
Determine whether the statements below are true or false.
(a) x ϵ {x}
(b) {x} ⊆ {x}
(c) {x} ϵ {x}
(d) {x} ϵ {{x}}
(e) {Ø} ⊆ {Ø,,{Ø}}
(f) Ø ⊆ {x}
(g) Ø ⊆ {x}
(a) x ϵ {x}
(b) {x} ⊆ {x}
(c) {x} ϵ {x}
(d) {x} ϵ {{x}}
(e) {Ø} ⊆ {Ø,,{Ø}}
(f) Ø ⊆ {x}
(g) Ø ⊆ {x}
Discrete Mathematics
show that 7(p->q) and p^7 q are logically equivalent without using truth table or using identities
Discrete Mathematics
7(pv(7p^q)) and 7p^7q are logically equivalence by developing a series of logical equivalence
