2.04 Deductive reasoning

In mathematics, we have postulates which are accepted facts. If we apply deductive reasoning to postulates and other starting hypotheses, we can construct logical arguments for new facts of mathematics.

Exploration

Consider the Venn diagram:

Consider whether each conclusion is valid or not. Explain your reasoning.

1. A polygon is a parallelogram. Therefore, it is not a trapezoid.

2. A polygon has two sides that are not parallel. Therefore, it is a parallelogram.

3. If a shape is a rhombus, then it must be a parallelogram.

There are serveral laws that can be used to construct an argument:

Recall the Law of Contrapositive:

The Law of Detachment says if the hypothesis of a true conditional statement is true, then the conclusion is also true.

For example:

Consider the number 11. Since p \to q is true and p is true since 11 is odd, by the law of detachment, 11 is the sum of an even number and an odd number.

The Law of Syllogism says if both p \to q \text{ and } q \to r are true, then p \to r is the logical conclusion."

For example:

Since p \to q is true and q \to r is true, by the law of syllogism, p \to r is true which means "If a shape has three sides, the angles sum to 360 degrees".

Deductive reasoning uses facts, definitions, and the laws of logic to form an argument. Inductive reasoning uses observations and patterns to form a conjecture. Deductive reasoning can prove an argument is true, while we can only use inductive reasoning to prove a conjecture is false.

Examples

Example 1

Determine the law of logic that is used in each argument.

a

If the sum of the angles in a polygon is 360 \degree, then it is a quadrilateral.

You see the following shape and know it is a quadrilateral.

Worked Solution

Create a strategy

Determine whether this represents the law of contrapositive, law of detachment, or law of syllogism.

Apply the idea

Let p \text{ and } q represent the following statements.

p	\text{The sum of the angles in a polygon is } 360 \degree.
q	\text{The shape is a quadrilateral.}

Since we are told that "if the sum of the angles in a polygon is 360 \degree, then it is a quadrilateral", we know that p \to q is true.

Also for the given figure the sum of the interior angles is 88\degree+85\degree+79\degree+108\degree=360\degree, so we can see that p is true. Using the law of detachment q must also be true, which states the shape is a quadrilateral.

Therefore, this is an example of the law of detachment.

Reflect and check

We can this symbolically:

Let p represent The sum of the angles in a polygon is 360\degree

Let q represent The shape is a quadrilateral.

\text{If the sum of the angles in a polygon is } 360 \degree \text{, then it is a quadrilateral}	p\to q
\text{The sum of the angles in the given polygon is } 360 \degree	p
\text{The given polygon is a quadrilateral}	q

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b

If you finish your homework, you will go to football game. If you go to the football game, you will sleep over at your friend's house.

If you finish your homework, then you will sleep over at your friends house.

Worked Solution

Create a strategy

Determine whether this represents the law of contrapositive, law of detachment, or law of syllogism.

Apply the idea

Let p, q, and r represent the following statements.

p	I finish my homework.
q	I go to the football game.
r	I sleep over at my friend's house.

So we can translate "If you finish your homework, you will go to football game" to p \to q and "If you go to the football game, you will sleep over at your friend's house" to q \to r.

Since we have p \to q \text{ and } q \to r, then we can say p \to r by the law of syllogism.

p \to r says that " If you finish your homework, then you will sleep over at your friends house."

Reflect and check

If the statement for p\to q or q\to r was false, then we would have a valid argument, but not necessarily a true statement as the result.

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Example 2

Which of the following conclusions can be drawn from these statements?

If we win the game, we will win the tournament. If we win the tournament, we will go out for pizza.

A

If we go out for pizza, we won the game.

B

If we win the game, we will go out for pizza

C

If we do not win the tournament, we will not go our for pizza.

Worked Solution

Create a strategy

Consider whether we can use the law of syllogism, law of contrapositive, or law of detachment to draw any of the conclusions.

Apply the idea

Let p represent We win the game

Let q represent We win the tournament.

Let q represent We go out for pizza.

\text {If we win the game, we will win the tournament.}	p\to q
\text{If we win the tournament, we will go out for pizza.}	q\to r
\text{Using the law of syllogism, we get:}\\ \text{If we win the game, we will go out for pizza }	p\to r

The law of syllogism confirms that the statement "If we win the game, we will go out for pizza" is true. So, the correct answer is B.

Reflect and check

Notice that A and C could be true, but aren't necessarily true. For choice A and C, we could go out for pizza even if we lose, so these do not follow from the original statements.

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Example 3

Write the logical argument symbolically, then determine whether the argument is valid. Justify using deductive reasoning.

If a quadrilateral has two sets of parallel sides, then it is a parallelogram. If a quadrilateral is a parallelogram, then its opposite angles are congruent. A quadrilateral does not have two sets of parallel sides. Therefore, its opposite angles are congruent.

Worked Solution

Create a strategy

Consider whether the law of syllogism, law of contrapositive, or law of detachment, or a few laws combined apply here.

Apply the idea

We can start by translating to symbols to see if the argument is valid.

Let p represent a quadrilateral has two set of parallel sides

Let q represent a quadrilateral is a parallelogram

Let r represent a quadrilateral has opposite angles that are congruent

Reason	Statement	Symbolic representation
\text{Given}	\text{If a quadrilateral has two sets of parallel sides,}\\ \text{then it is a parallelogram.}	p \to q
\text{Given}	\text {If a quadrilateral is a parallelogram, }\\ \text{then its opposite angles are congruent. }	q \to r
\text{Law of syllogism}	\text {If a quadrilateral has two sets of parallel sides, }\\ \text{then its opposite angles are congruent. }	p \to r
\text{Given}	\text {A quadrilateral does not have two sets of parallel sides}	\sim p
\text{Invalid conclusion}	\text{Therefore, its opposite angles are congruent.}	\therefore r

The first and second statements are true, and the third statement can be created using the law of syllogism. However, the conclusion contradicts the law of detachment, making the argument invalid and the conclusion false.

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In mathematics, we have postulates which are accepted facts. If we apply deductive reasoning to postulates and other starting hypotheses, we can construct logical arguments for new facts of mathematics. A new fact is called a theorem and its argument is called its proof.

We can use deductive reasoning to write proofs in a variety of ways. Formal proofs have a more rigid structure and can be helpful when we want to present our justification in a very organized way.

The structure of a two column proof follows:

Every statement in a proof must be show to be true. In geometry, an undefined term is a term or word that does not require further explanation or description. The include a point, set, line, and plane. A defined term is a term that has a formal definition and can be defined using other geometrical terms.

We have learned many properties and laws that can be used to make arguments. Recall the properties of equality:

We also have the properties of congruence:

Properties of segment congruence tell us that segment congruence is reflexive, symmetric, and transitive

Properties of angle congruence tell us that angle congruence is reflexive, symmetric, and transitive

Examples

Example 4

Consider the vertical angles theorem, which states that vertical angles are congruent.

Use transformations to prove the following:

Given: \angle ABC and \angle DBE are vertical angles

Prove: \angle ABC \cong \angle DBE

Worked Solution

Create a strategy

We can use translations, reflections, rotations, and dilations to demonstrate congruence between figures.

Apply the idea

We are given that \angle ABC and \angle DBE are vertical angles. Since \angle ABC and \angle DBE are vertical angles, they share vertex B. We can use tracing paper to rotate \angle ABC clockwise 180\degree around vertex B and verify that its image is \angle DBE. Therefore, \angle ABC \cong \angle DBE.

Reflect and check

This is an example of an informal paragraph proof that uses an annotated diagram.

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Example 5

Consider the congruent supplements theorem, which states that if two angles are supplementary to the same angle then they are congruent.

Construct a two column proof to prove the following:

Given:

• \angle A and \angle C are supplementary

• \angle B and \angle C are supplementary

Prove: \angle A \cong \angle B

Worked Solution

Apply the idea

To prove: \angle A \cong \angle B

	Statements	Reasons
1.	\angle A and \angle C are supplementary	Given
2.	\angle B and \angle C are supplementary	Given
3.	m\angle A +m\angle C=180\degree	Definition of supplementary angles
4.	m\angle B +m\angle C=180\degree	Definition of supplementary angles
5.	m\angle A +m\angle C=m\angle B +m\angle C	Transitive property of equality
6.	m\angle A=m\angle B	Addition property of equality
7.	\angle A \cong \angle B	Definition of congruent angles

Reflect and check

We can see that the proofs for the congruent supplements theorem and the congruent complements theorem are very similar. The flow chart proof and two column proof use the same steps but display them differently.

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