Geometry Help: Deductive Reasoning

Discovering the laws of Detachment and Syllogism and learning how to incorporate them into what we know.

Law of Detachment

If P→Q is a true statement and P is true, then Q is true.

Example 1: What can you conclude?

A) If you finish your dinner, you may have a pieceof cake. You eat all your dinner.

Now P would be finishing dinner. The problem states you do, so P is true. We can thus infer that Q is true also.

Conclusion-got cake

B) If it is snowing outside, it must be below freezing. The temperature is 20 degrees F.

Don’t let this one trick you. Mind your P’s and Q’s. P talks about snowing, but they never tell you if it’s snowing. Can it be 20°F without snowing? Of course. So we can draw no conclusion.

-No conclusion

C)If an angle is obtuse, then it is not acute. Angle ABC is not obtuse.

This one is also tricky. P says ‘if an angle is obtuse, while the clue says ‘is not obtuse’. We can draw no conclusion.

-No conclusion

D) If a figure is a rectangle, then it has two pairs of parallel lines. Figure ABCD is a square.

Is a square a rectangle? Always. So we can draw our conclusion.

Conclusion-Figure ABCD has two pairs of parallel lines.

Law of Syllogism

If p→q and q→r, then p→r is a true statement.

In the law of Syllogism, the one statement will have P and Q; the other will have Q and R. They aren’t always in order; they could be flip-flopped. How do you tell them apart? By Q. Notice Q is in both statements. The one with Q at the end is P→Q. The one with Q in the front is Q→R. Once you figure this out, you can state P→R.

Example 2

If you are doing algebra, then you are doing math. If you are solving mathematical equations, then you are doing algebra. Find the true statement.

Now before you get carried away (or run away), look at the sentences and pick out witch statement is stated twice. The algebra one, right. That’s our Q. Let’s label all of the phrases.

  • P- solving mathematical equations
  • Q- doing algebra
  • R- doing math

Now it’s simple. State P and R in an ‘if-then’ statement. If you don’t know what this means, see Geometry Help: Conditional Statements.

Answer – If you are solving mathematical equations, you are doing math.

Further Reading

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