Give some examples of contraposition?

Question

Give some examples of contraposition?

Answer 1

Examples

Take the statement "All red things have color." This can be equivalently expressed as "If an object is red, then it has color."

The contrapositive is "If an object does not have color, then it is not red". This follows logically from our initial statement and, like it, it is evidently true.

The inverse is "If an object is not red, then it does not have color." Again, an object which is blue is not red, and still has color. Therefore the inverse is also false.

The converse is "If an object has color, then it is red." Objects can have other colors, of course, so, the converse of our statement is false.

The contradiction is "There exists a shade of red that does not have the properties of color". If the contradiction were true, then both the converse and the inverse would be correct in exactly that case where the shade of red is not a color. However, in our world this statement is entirely untrue (and therefore false).

In other words, the contrapositive is logically equivalent to a given conditional statement, though not sufficient for a biconditional.

Similarly, take the statement "All quadrilaterals have four sides," or equivalently expressed "If a shape is a quadrilateral, then it has four sides."

The contrapositive is "If a shape does not have four sides, it is not a quadrilateral." This follows logically, and as a rule, contrapositives share the truth value of their conditional.

The inverse is "If a shape is not a quadrilateral, then it does not have four sides." In this case, unlike the last example, the inverse of the argument is also true.

The converse is "If a shape has four sides, then it is a quadrilateral." Again, in this case, unlike the last example, the converse of the argument is true.

Since the statement, and the converse are both true, it is called a biconditional, and can be expressed as "A shape is a quadrilateral if, and only if, it has four sides." That is, having four sides is both necessary to be a quadrilateral, and alone sufficient to deem it a quadrilateral.