Deduction is a type of reasoning that moves from the top down: it starts with a general theory, then relates it to a specific example. We start with a broad statement that we know to be true, and then we apply it to a particular situation.
It's like the Ten Commandments. No, really; hear us out.
Number eight says "do not steal," which is a general statement: stealing is bad, in general. We could then use deductive reasoning to apply that statement to a specific example: "I probably shouldn't steal this particular Xbox from my best friend's bedroom after knocking him out, right?" Right. That's an example of deductive reasoning: we applied the general theory (stealing is bad) to the particular case (therefore, stealing your friend's Xbox is also bad).
Good news: you already know how to prove stuff deductively. Most of the algebraic and geometric proofs you've done so far have been deductive proofs. We start with some kind of general rule, like "supplementary angles always add up to 180°," and apply it to a specific example, like "angle 1 has a measure of 75°, so an angle supplementary to angle 1 must have a measure of 105°."
With deductive proofs, we usually use postulates and theorems as our general statements and apply 'em to specific examples. And we usually do this a bunch of different times in a single proof.
For example, take a gander at the following formal proof.
Given: w = x, x = y, y = z
Prove: w = z
| Statements | Reasons |
| 1. w = x | Given |
| 2. x = y | Given |
| 3. y = z | Given |
| 4. w = y | Transitive Property (1 and 2) |
| 5. w = z | Transitive Property (4 and 3) |
Check out Statement #4. We used the transitive property as our general theory, since we know it's always true. Then we used deductive reasoning to apply the transitive property to our specific example: if w = x and x = y, then it must be true that w = y.
After that, we used deduction again in Statement #5. We can stack up mini-deductions like this, over and over, until we finally prove whatever we set out to prove. It's like building a LEGO set out of mind-bricks.
Example 1
Is the following situation an example of deductive reasoning? Why or why not? The area of any circle is given by the formula A = πr2. Therefore, the area of a circle with a radius of 5 cm is 25π cm2. |
Example 2
Is the following situation an example of deductive reasoning? Why or why not? Marceline carefully measures the internal angles of 500 different rectangles, and she finds that every time, the angles add up to 360°. Therefore, she concludes that all rectangles must have internal angles that add up to 360°. |
Example 3
What general theory or idea can we use to deductively prove Statement #3 in the following formal proof? Given: A = B and X = Y
|
Exercise 1
What general theory can we use to deductively prove Statement #6 in the following formal proof?
Given: Q = R and S = TProve: QS + RT = Q(S + T)
| 1. Q = R | Given |
| 2. S = T | Given |
| 3. S + T = T + T | Addition Property (2) |
| 4. S + T = S + T | Substitution Property (2 and 3) |
| 5. Q(S + T) = Q(S + T) | Multiplication Property (4) |
| 6. QS + QT = Q(S + T) | ? |
| 7. QS + RT = Q(S + T) | ? |
Exercise 2
What general theory can we use to deductively prove Statement #7 in the following formal proof?
Given: Q = R and S = TProve: QS + RT = Q(S + T)
| 1. Q = R | Given |
| 2. S = T | Given |
| 3. S + T = T + T + T | Addition Property (2) |
| 4. S + T = S + T | Substitution Property (2 and 3) |
| 5. Q(S + T) = Q(S + T) | Multiplication Property (4) |
| 6. QS + QT = Q(S + T) | ? |
| 7. QS + RT = Q(S + T) | ? |
Exercise 3
Donald measures the area of his triangular backyard, along with the areas of several of his neighbor's yards, which are also triangular (he lives in a weird neighborhood). He discovers that the area of each yard is consistently half of its base times its height, so he uses that info to come up with an area formula he believes will work for any triangle in the world: 
. Did he use deductive reasoning to reach his conclusion?
Exercise 4
Veronica is trying to find the roots of the quadratic equation x2 + 8x – 11 = 0 for her math homework. She looks up the quadratic formula on Shmoop, then plugs her coefficients into the formula and gets x ≈ -9.196 and x ≈ 1.196. Did she use deductive reasoning to get her answer?