Science:Infinite Series Module/Appendices/The Contrapositive and the Divergence Test/Contrapositive Examples
British Columbia and Canada
You are probably familiar with at least a little Canadian geography, and know that British Columbia (BC) is a geographical area within Canada. BC is a province belonging to Canada, as shown in the diagram below.
Consider the statement
If I am standing in BC, then I am standing in Canada.
The contrapositive of this example is
If I am not standing in Canada, then I am not standing in BC.
The contrapositive is certainly true because the entire province of BC is a part of Canada. In fact, the contrapositive is true because the original statement is true: if a part of BC were not in Canada, then both the original statement and the contrapositive would be false.
Trigonometric Function
Consider the statement
If x is equal to zero, then sin(x) is equal to zero.
This statement is certainly true, and its contrapositive is
If sin(x) is not zero, then x is not zero.
Again, the contrapositive is certainly true. If we take x to be any value so that is not zero, then x cannot be zero.
If the Grass is Not Wet
Suppose we are in an open field of grass.
Consider the statement
If it is raining, then the grass is wet.
The contrapositive of this example is
If the grass is not wet, then it is not raining.
Sure, the grass could get wet if we were watering the grass. But if the grass is not wet, it can't be raining. Otherwise, the grass would be wet.
Equivalence
At this point, it may not be clear that in each of the above cases,
- the contrapositive was true because the original statement was true
- the given statement and its contrapositive are equivalent
To get a sense of why this would be so, the next example takes a closer look at the contrapositive.