Question

Which statement is logically equivalent to the conditional statement? "it is not true that the car is red or the house is not brown." responses a "the car is red and the house is brown.""the car is red and the house is brown." b "the car is red and the house is not brown.""the car is red and the house is not brown." c "the car is not red and the house is brown.""the car is not red and the house is brown." // d "the car is not red and the house is not brown."

Answer 1

Let's break this problem down into logical components.

First, we have a statement which says "it is not true that the car is red or the house is not brown."

We will split it into two constituents:
- P: The car is red
- Q: The house is not brown

In logic, "or" refers to inclusive-or, meaning that if either P or Q is true, then "P or Q" is true. Therefore "it is not true that P or Q" = "not (P or Q)". To simplify it, we apply De Morgan's law, which provides the equivalence "not (P or Q)" = "not P and not Q".

Here, "not P" would be "the car is not red", and "not Q" would be "the house is brown" (Which is the negation of "the house is not brown").

Thus, the logical equivalent of the given statement is "the car is not red and the house is brown". Consequently, among the options you have given, option c, namely "the car is not red and the house is brown" would be correct.