Determine which if any of the three statements are equivalent. Give a reason for your conclusion.

1) If the house is not white,then the house is not small.
2) If house is not white and the house is not small
3)It is false that the house is white or
that the house is not small.

None of the statements are equivalent.

#1 is an if.. then statement
#2 is an if..and set of conditions, not a statement
#3 is two statements about being white or small, neither of which are necessarily true

To determine if any of the three statements are equivalent, let's analyze each statement and compare them to each other.

1) "If the house is not white, then the house is not small."
This statement implies that if the house is not white, it cannot be small. In other words, it affirms that the house's size is dependent on whether it is white or not.

2) "If the house is not white and the house is not small."
This statement states that both conditions must be true: The house must not be white, and it must not be small. So the statement implies that if either of these conditions is not true, then the overall statement is false.

3) "It is false that the house is white or that the house is not small."
This statement suggests that either the house is not white or it is not small, but not both. In other words, it denies the possibility of both conditions existing simultaneously.

Now, let's compare these statements:

Statement 1 implies that if a house is not white, it is not small. It does not make any assertions about the possibility of the house being both not white and not small.

Statement 2 implies that both conditions of the house not being white and not being small must be true simultaneously.

Statement 3 denies the possibility of the house being either white or not small.

Based on this analysis, we can conclude that statements 1) and 3) are equivalent. This is because they both deny the possibility of the house being white or not small. Instead, they assert that the house must be either white and small, or not white and not small.

Statement 2) is different from the other two statements because it requires both conditions of not being white and not being small to be true simultaneously.