Truth Tables - conditional statement.?

Question

I’m doing homeschooling and need to submit this within this week. So, please help. Thanks


2. Let a, b, c be integers. Consider the following conditional statement. -----------------------
If a divides bc, then a divides b or a divides c. --------------------------------------------
Which of the following statements have the same meaning as this conditional statement and which ones are negations of this conditional statement?
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Note:
This is not asking which statements are true and which are false. It is asking which statement are logically equivalent to the given statement. It might be helpful to let P represent the hypothesis of the given statement, Q represent the conclusion, and then determine a symbolic representation for each statement. Instead of using truth tables, try to use already established logical equivalencies to justify your conclusions.

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1-If a divides b or a divides c, then a divides bc.
2-If a does not divide b or a does not divide c, then a does not divide bc.
3-a divides bc, a does not divide b, and a does not divide c.
4-If a does not divide b and a does not divide c, then a does not divide bc.
5-a does not divide bc or a divides b or a divides c.
6-if a divides bc and a does not divide c, then a divides b.
7-if a divides bc or a does not divide b, then a divides c.

So confusing, but I LOVE YOU GUYS FOR HELPING ME!!!!!!!

Misty

Answer 1

Choice 4 is the contrapositive, which is equivalent to the original statement. Note that the negation of the statement's consequent is the new precedent and vice versa. Also, note the use of DeMorgan's law, which changes OR to AND.
I think you can also derive 5 and 6, depending on what axiom schema you are using.

Answer 2

2. Let's assign letters to statements.
D = "a divides bc"
B = "a divides b"
C = "a divides c"

Your equivalent statement in symbolic logic is:

D -> (B v C)

I'm going to figure out the negation only (as the answer to the first question involves truth tables)

So we want to convert

~( D -> (B v C))

We can convert the conditional to a disjunction by negating the antecedent.

~(~D v (B v C))

By DeMorgan's law,

~~D . ~(B v C)

And by DeMorgan's law again,

~~D . ~B . ~C

By double negation,

D . ~B . ~C

Translation: "A divides bc AND A does NOT divide B AND A does NOT divide C." So the answer is #3.

Answer 3

statement 2 is the contrapositive to the conditional statement. I'm not sure if this will help but it is known that if the conditional statement is true then the contrapositive is true. And if the conditional statement is false then the contrapositive is false. To me, this sounds like both statements are logically equivalent