By Courtney Taylor. Statistics Expert
Courtney K. Taylor, Ph.D. is an associate professor of mathematics at Anderson University in Anderson, Indiana. He teaches a wide variety of courses throughout mathematics, including those involving statistics.
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We will start by examining logical statements. In logic, a statement is a sentence that is either true or false. Statements can concern anything, as long as we can assign a truth value to the sentence, but in mathematics statements typically concern mathematical objects.
Examples of statements include:
- The sum of the angles of a triangle is 180 degrees.
- There is a Ferrari in my garage.
- It is 80 degrees outside today.
- Dogs are better than cats.
- It is a nice day today.
Formation of Conditional Statements
Just as in set theory we use certain sets, such as the empty set and set operations to form new sets, we can use statements and logical forms to construct new statements. We will let P and Q be logical statements. This means that either of these letters can be replaced by any sentence which is either true or false. A conditional statement is of the form “If P, then Q.”
Examples of conditional statements abound:
- If a triangle is a right triangle, then the square of its hypotenuse equals the sum of the squares of the other two sides.
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- If 2+9 =11, then 9+2=11
- If candidate A receives
270 electoral votes, then candidate A is the winner of the election.
Truth of Conditional Statements
- P and Q are both true. In this case the conditional statement “If P then Q ” is true.
- P is true and Q is false. In this case the conditional statement “If P then Q ” is false. The reason for this is relatively straightforward. When P is true, then this implies that Q must also be true in the conditional statement. Since Q is false, the entire conditional is false.
- P is false and Q is true. In this case the conditional statement “If P then Q ” is true. This is a little tricky to understand. Probably the best way of thinking about this is that the conditional is true automatically, or vacuously true when P is false. For instance, we will analyze the conditional “If it rains today, then I will bring an umbrella to my office.” If it doesn’t rain today, then it doesn’t matter if I bring an umbrella or not. The statement is only false if it rains today and I don’t bring an umbrella to work.
- P is false and Q is false. In this case the conditional statement “If P then Q ” is true. This is true for the same reason as the third conditional statement in our list.