Two statements must have the same meaning in order to be logically equivalent to each other.
H
Practice makes perfect
Two statements are logically equivalent when they have the same truth value. This means that the two statements can be true in their own contexts, but they just have to have the same meaning. The given statement is a conditional statement in the form of if p then q, where p is called the hypothesis and q is the conclusion.
p ⇒ q
If the hypothesis is true, then the conclusion must be true. If the conclusion is false, the hypothesis cannot be true. However, if the conclusion is true, the hypothesis may or may not be true. If the hypothesis is false, the conclusion may or may not be false. Let's look at the given statement.
If a figure is a rhombus, then it has four sides.
A figure is a rhombus is the hypothesis and has four sides is the conclusion. That means if a figure is a rhombus, then it must have four sides. Let's evaluate the statements in options F, G, H, and I to determine which of them is logically equivalent to the original statement.
Option F
Let's look at the statement in option F.
If a figure is not a rhombus, then it does not have four sides.
In reference to the original statement, this statement is arguing if the hypothesis is false, then the conclusion is false. However, this is not logically equivalent to the original statement, because a false hypothesis means the conclusion may or may not be false. For example, a rectangle is not a rhombus, but it does have four sides.
Option G
Let's look at the statement in option G.
If a figure is a rectangle, then it has four sides.
This statement is not logically equivalent to the original statement because a rectangle is not a rhombus. To be a rhombus, all sides must be equal. However, the sides of a figure do not need to be equal to qualify as a rectangle. Therefore, the two statements do not have the same truth value.
Option H
Let's look at the statement in option H.
If a figure does not have four sides, then it is not a rhombus.
In reference to the original statement, this statement is arguing if the conclusion is false, then the hypothesis is false. This is logically equivalent to the original statement. The original statement is stating that in order to be a rhombus, the figure must have four sides. Therefore, if the figure does not have four sides, it cannot be a rhombus.
Option I
Let's look at the statement in option I.
If a figure has four sides, then it could be a rhombus.
In reference to the original statement, this statement is arguing if the conclusion is true, then the hypothesis may or may not be true. While this is a correct assumption, it is not telling us the same definite fact as the original statement. Therefore, the two statements do not hold the same truth value and are not logically equivalent. The correct answer is option H.
