Logic and Proof Exercises

Example 1

What general theory can we use to deductively prove Statement #6 in the following formal proof?

Given: Q = R and S = T
Prove: 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)	?

Hint

We're making those parentheses on the left vanish into thin air. Which property lets us do that?

Example 2

What general theory can we use to deductively prove Statement #7 in the following formal proof?

Given: Q = R and S = T
Prove: 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)	?

Hint

One letter changes between Statement #6 and 7.

Example 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?

Hint

Did he move from general to specific?

Example 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?

Hint

Did she apply a general theory to something specific, or vice versa?

Logic and Proof Exercises

Example 1

Example 2

Example 3

Example 4

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