Which statement best explains the relationship between numbers divisible by 9 and 3. A. A Number that is divisible by 9 is also divisible by 3 because 3is a factor of 9. B. A Number that is divisible by 9 is also disvisible by 3 because all factors of 9 are also factor of 3. C. Some of the numbers that are divisible by 9 are also divisible by 3. D. Numbers that are divisible by 9 are never divisible by 3

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fdelacruz19

09.09.2021

Mathematics

We start by applying De Morgan's law:

$\neg (p \wedge q) \vee p \equiv [(\neg p) \vee (\neg q)] \vee p.$

We now use the fact that the disjunction is commutative:

$[(\neg p) \vee (\neg q)] \vee p \equiv p \vee [(\neg p) \vee (\neg q)].$

We now use the fact that the disjunction is distributive:

$p \vee [(\neg p) \vee (\neg q)] \equiv [p \vee (\neg p)] \vee (\neg q).$

Using the law of excluded middle, we have $p \vee (\neg p) \equiv \textrm{TRUE}$ , so that:

$[p \vee (\neg p)] \vee (\neg q) \equiv \textrm{TRUE} \vee (\neg q).$

We finally use the fact that TRUE is the absorbing element of the disjunction:

$\textrm{TRUE} \vee (\neg q) \equiv \textrm{TRUE}.$

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