Q:

How to know if a conditional independence is true or false

I'm learning Bayesian networks and I have to "guess" if the following conditional independence are true or false using the following table:

And the conditional independence are: 

$I_p(A, B)$
$I_p(A, C)$

p is the probability distribution.
A = a, B = b, and C = c.
I have no idea about how to know or which operation do I have to do to know if those conditional independence are true or false.
What do I have to do to know if those conditional independence are true or false?

A:

It appears that both from your book, and Learning Bayesian Networks by Richard Neapolitan uses the same notations. I've never come across with the notation $I_P(..)$ while dealing with Bayesian Networks, but it turns out your book is not the only one. This notation simply means Independence in P (Neapolitan page ~29). For example, if $P(A|B)=P(A)$, then $I_P(A,B)$, which means $A$ and $B$ are independent. If $P(A|B,C)=P(A|C)$ (for all $a,b,c$), then $I_P(A,B|C)$, which means $A$ and $B$ are independent conditioned on C. 
In your question, you ask for the conditional independences $I_P(A,B)$ and $I_P(A,C)$, but according to the definitions there ask for usual independence, not conditional, which can still be referred as conditional independence on nothing. 
Anyway, either it is conditional or not, we have the full joint distribution and we can calculate any probability we want using the table. For example, a marginal and a two-variable joint can be calculated as follows: $$P(A=a)=\sum_b\sum_c{P(A=a,B=b,C=c)}$$ $$P(A=a,B=b)=\sum_c{P(A=a,B=b,C=c)}$$
A conditional probability can be calculated using Bayes Law:
$$P(A=a|B=b)=\frac{P(A=a,B=b)}{P(B=b)}$$
$$P(A=a,B=b|C=c)=\frac{P(A=a,B=b,C=c)}{P(C=c)}$$
You just need to plug in the numbers and verify that $P(A,B)=P(A)P(B)$ for $I_P(A,B)$ for the first case, assuming it questions independence of $A$ and $B$ conditioned on nothing, based on the definitions in both books.

