Discrete Mathematics Answers

State the Dijkstra’s algorithm for a directed weighted graph with all non-negative edge weights

If A={2,4,6}and B={1,3,5}. Then \\(A\\cup B\\) is equal to__________
a.{1,2,3,4,6}
b.{0,2,3,4,5,6}
c.{1,2,4,5,6}
d.{1,2,3,4,5,6}

1/ Find weather the two function are invertible or not, if it is find out its inverse (f^1(x))

1. f:[-π/2, π/2]→[-1,1]; f(x)=sin x
2. f:[0,π]→[-2,2]; f(x)=2cos x

2/ The function f(x)=5/9(x-32) converts Fahrenheit temperatures into Celsius, what is the opposite function for the opposite conversion?

Let ρ ⊆ S × T, for finite sets S and T. Define fρ : S −→ P(T) be such that fρ(a) = {b | aρb}.
a) Prove that ρ is reflexive if and only if a ∈ fρ(a), for every a ∈ S.
b) Prove that ρ is symmetric if and only if a ∈ fρ(b), for every a ∈ S and b ∈ fρ(a).
c) Prove that ρ is transitive if and only if fρ(b) ⊆ fρ(a), for every a ∈ S and b ∈ fρ(a).
d) Prove that ρ is not many-to-one if and only if fρ(a) ∩ fρ(b) 6= ∅ implies a = b, for every
a, b ∈ S.
e) Can fρ be onto? Explain your reasoning.

Formulate corresponding proof principles to prove the following properties about defined sets

1. A=B⇔A⊆B and B ⊆ A
2. De Morgan’s Law by mathematical induction
3. Laws for three non-empty finite sets A, B, and C

function f(x) = 5/9(x-32) converts Fahrenheit temperatures into Celsius. What is the function for opposite conversion?

Find out if the following functions are invertible or not, If it is invertible, then find the rule of the inverse (f^(-1) (x))

1.
f:k → k^+
f(x)=x^2

2.
k^+ → k^+
f(x)=1/x

3.
f:k^+ → k^+
f(x)=x^2

Let A,B,C be subsets of a set. Prove that A ∩ B ⊆ C iff A⊆B' U C

Give a self conjugate partition of 21, with justification.

Show that ( pimpliesq)^(qimplies~ p) equivalent~ p
i) with truth table.
ii) without truth table.