Let a, b, and 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, which ones are the negations, and which ones are not neither? Justify your answers using logical equivalences or truth tables.
A) If a does not divide b or a does not divide c, then a does not divide bc.
B) If a does not divide b and a does not divide c, then a does not divide bc.
C) If a divides bc and a does not divide c, then a divides b.
D) If a divides bc or a does not divide b, then a divides c. (e) a divides bc, a does not divide b, and a does not divide c.

Step-by-step explanation:

Given that the logical statement is

"If a divides bc, then a divides b or a divides c"

we can see that a must divide one either b or c from the statement above

A) If a does not divide b or a does not divide c, then a does not divide bc.

This is False because a can divide b or c

 B) If a does not divide b and a does not divide c, then a does not divide bc.

this is True for a to divide bc it must divide b or c (either b or c)

C) If a divides bc and a does not divide c, then a divides b.

This is True since a can divide bc and it cannot divide c, it must definitely divide b

D) If a divides bc or a does not divide b, then a divides c.

This is True since a can divide bc and it cannot divide b, it must definitely divide c

E) a divides bc, a does not divide b, and a does not divide c.

This is False for a to divide bc it must divide one of  b or c

Statement A is not the same as the original statement.

Statement B is the negation of the original statement.

Statement C is the same as the original statement.

Statement D is not the same as the original statement.

Condition E is not a statement, but a set of conditions without any logical implications.

Given that;

The conditional statement:

If a divides bc, then a divides b or a divides c

A) If a does not divide b or a does not divide c, then a does not divide bc.

This statement is not the same as the original conditional statement.

The original statement states that if a divides bc, then a divides b or a divides c.

However, statement A states the opposite - if a does not divide b or a does not divide c, then a does not divide bc.

So, this is not the same as the original statement.

B) If a does not divide b and a does not divide c, then a does not divide bc.

This statement is actually the negation of the original conditional statement.

The original statement states that if a divides bc, then a divides b or a divides c.

The negation of this statement would be that if a does not divide b and a does not divide c, then a does not divide bc.

So, statement B is the negation of the original statement.

C) If a divides bc and a does not divide c, then a divides b.

This statement is the same as the original conditional statement. It states that if a divides bc and a does not divide c, then a divides b.

This is equivalent to the original statement, which states that if a divides bc, then a divides b or a divides c.

D) If a divides bc or a does not divide b, then a divides c.

This statement is not the same as the original conditional statement.

The original statement states that if a divides bc, then a divides b or a divides c.

However, statement D states that if a divides bc or a does not divide b, then a divides c.

This is a different condition altogether, so it is not equivalent to the original statement.

E) a divides bc, a does not divide b, and a does not divide c.

This is not a statement but rather an additional condition specified.

It describes a scenario where a divides bc, a does not divide b, and a does not divide c.

However, it doesn't provide any logical implications or conclusions like the conditional statements we have been discussing.

Therefore, we get;

Statement A is not the same as the original statement.

Statement B is the negation of the original statement.

Statement C is the same as the original statement.

Statement D is not the same as the original statement.

Condition E is not a statement, but a set of conditions without any logical implications.

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