- Algebra
- Arithmetic
- Whole Numbers
- Numbers
- Types of Numbers
- Odd and Even Numbers
- Prime & Composite Numbers
- Sieve of Eratosthenes
- Number Properties
- Commutative Property
- Associative Property
- Identity Property
- Distributive Property
- Order of Operations
- Rounding Numbers
- Absolute Value
- Number Sequences
- Factors & Multiples
- Prime Factorization
- Greatest Common Factor
- Least Common Multiple
- Squares & Perfect Squares
- Square Roots
- Squares & Square Roots
- Simplifying Square Roots
- Simplifying Radicals
- Radicals that have Fractions
- Multiplying Radicals

- Integers
- Fractions
- Introducing Fractions
- Converting Fractions
- Comparing Fractions
- Ordering Fractions
- Equivalent Fractions
- Reducing Fractions
- Adding Fractions
- Subtracting Fractions
- Multiplying Fractions
- Reciprocals
- Dividing Fractions
- Adding Mixed Numbers
- Subtracting Mixed Numbers
- Multiplying Mixed Numbers
- Dividing Mixed Numbers
- Complex Fractions
- Fractions to Decimals

- Decimals
- Exponents
- Percent
- Scientific Notation
- Proportions
- Equality
- Properties of equality
- Addition property of equality
- Transitive property of equality
- Subtraction property of equality
- Multiplication property of equality
- Division property of equality
- Symmetric property of equality
- Reflexive property of equality
- Substitution property of equality
- Distributive property of equality

- Commercial Math

- Calculus
- Differential Calculus
- Limits calculus
- Mean value theorem
- L’Hôpital’s rule
- Newton’s method
- Derivative calculus
- Power rule
- Sum rule
- Difference rule
- Product rule
- Quotient rule
- Chain rule
- Derivative rules
- Trigonometric derivatives
- Inverse trig derivatives
- Trigonometric substitution
- Derivative of arctan
- Derivative of secx
- Derivative of csc
- Derivative of cotx
- Exponential derivative
- Derivative of ln
- Implicit differentiation
- Critical numbers
- Derivative test
- Concavity calculus
- Related rates
- Curve sketching
- Asymptote
- Hyperbolic functions
- Absolute maximum
- Absolute minimum

- Integral Calculus
- Fundamental theorem of calculus
- Approximating integrals
- Riemann sum
- Integral properties
- Antiderivative
- Integral calculus
- Improper integrals
- Integration by parts
- Partial fractions
- Area under the curve
- Area between two curves
- Center of mass
- Work calculus
- Integrating exponential functions
- Integration of hyperbolic functions
- Integrals of inverse trig functions
- Disk method
- Washer method
- Shell method

- Sequences, Series & Tests
- Parametric Curves & Polar Coordinates
- Multivariable Calculus
- 3d coordinate system
- Vector calculus
- Vectors equation of a line
- Equation of a plane
- Intersection of line and plane
- Quadric surfaces
- Spherical coordinates
- Cylindrical coordinates
- Vector function
- Derivatives of vectors
- Length of a vector
- Partial derivatives
- Tangent plane
- Directional derivative
- Lagrange multipliers
- Double integrals
- Iterated integral
- Double integrals in polar coordinates
- Triple integral
- Change of variables in multiple integrals
- Vector fields
- Line integral
- Fundamental theorem for line integrals
- Green’s theorem
- Curl vector field
- Surface integral
- Divergence of a vector field
- Differential equations
- Exact equations
- Integrating factor
- First order linear differential equation
- Second order homogeneous differential equation
- Non homogeneous differential equation
- Homogeneous differential equation
- Characteristic equations
- Laplace transform
- Inverse laplace transform
- Dirac delta function

- Differential Calculus
- Matrices
- Pre-Calculus
- Lines & Planes
- Functions
- Domain of a function
- Transformation Of Graph
- Polynomials
- Graphs of rational functions
- Limits of a function
- Complex Numbers
- Exponential Function
- Logarithmic Function
- Sequences
- Conic Sections
- Series
- Mathematical induction
- Probability
- Advanced Trigonometry
- Vectors
- Polar coordinates

- Probability
- Geometry
- Angles
- Triangles
- Types of Triangles
- Special Right Triangles
- 3 4 5 Triangle
- 45 45 90 Triangle
- 30 60 90 Triangle
- Area of Triangle
- Pythagorean Theorem
- Pythagorean Triples
- Congruent Triangles
- Hypotenuse Leg (HL)
- Similar Triangles
- Triangle Inequality
- Triangle Sum Theorem
- Exterior Angle Theorem
- Angles of a Triangle
- Law of Sines or Sine Rule
- Law of Cosines or Cosine Rule

- Polygons
- Circles
- Circle Theorems
- Solid Geometry
- Volume of Cubes
- Volume of Rectangular Prisms
- Volume of Prisms
- Volume of Cylinders
- Volume of Spheres
- Volume of Cones
- Volume of Pyramids
- Volume of Solids
- Surface Area of a Cube
- Surface Area of a Cuboid
- Surface Area of a Prism
- Surface Area of a Cylinder
- Surface Area of a Cone
- Surface Area of a Sphere
- Surface Area of a Pyramid
- Geometric Nets
- Surface Area of Solids

- Coordinate Geometry and Graphs
- Coordinate Geometry
- Coordinate Plane
- Slope of a Line
- Equation of a Line
- Forms of Linear Equations
- Slopes of Parallel and Perpendicular Lines
- Graphing Linear Equations
- Midpoint Formula
- Distance Formula
- Graphing Inequalities
- Linear Programming
- Graphing Quadratic Functions
- Graphing Cubic Functions
- Graphing Exponential Functions
- Graphing Reciprocal Functions

- Geometric Constructions
- Geometric Construction
- Construct a Line Segment
- Construct Perpendicular Bisector
- Construct a Perpendicular Line
- Construct Parallel Lines
- Construct a 60° Angle
- Construct an Angle Bisector
- Construct a 30° Angle
- Construct a 45° Angle
- Construct a Triangle
- Construct a Parallelogram
- Construct a Square
- Construct a Rectangle
- Locus of a Moving Point

- Geometric Transformations

- Sets & Set Theory
- Statistics
- Collecting and Summarizing Data
- Common Ways to Describe Data
- Different Ways to Represent Data
- Frequency Tables
- Cumulative Frequency
- Advance Statistics
- Sample mean
- Population mean
- Sample variance
- Standard deviation
- Random variable
- Probability density function
- Binomial distribution
- Expected value
- Poisson distribution
- Normal distribution
- Bernoulli distribution
- Z-score
- Bayes theorem
- Normal probability plot
- Chi square
- Anova test
- Central limit theorem
- Sampling distribution
- Logistic equation
- Chebyshev’s theorem

- Difference
- Correlation Coefficient
- Tautology
- Relative Frequency
- Frequency Distribution
- Dot Plot
- Сonditional Statement
- Converse Statement
- Law of Syllogism
- Counterexample
- Least Squares
- Law of Detachment
- Scatter Plot
- Linear Graph
- Arithmetic Mean
- Measures of Central Tendency
- Discrete Data
- Weighted Average
- Summary Statistics
- Interquartile Range
- Categorical Data

- Trigonometry
- Vectors
- Multiplication Charts
- Time Table
- 2 times table
- 3 times table
- 4 times table
- 5 times table
- 6 times table
- 7 times table
- 8 times table
- 9 times table
- 10 times table
- 11 times table
- 12 times table
- 13 times table
- 14 times table
- 15 times table
- 16 times table
- 17 times table
- 18 times table
- 19 times table
- 20 times table
- 21 times table
- 22 times table
- 23 times table
- 24 times table

- Time Table

# Law of Detachment – Explanation and Examples

*The law of detachment states that if the antecedent of a true conditional statement is true, then the consequence of the conditional statement is also true.*

This law regards the truth value of conditional statements.

Before moving on with this section, make sure to review conditional statements and the law of syllogism.

This section covers:

**What is the Law of Detachment?****Law of Detachment Examples**

## What Is the Law of Detachment?

The law of detachment states that if a conditional statement is true and its antecedent is true, then the consequence must also be true.

Recall that the antecedent is what follows the word “if” in a conditional statement. A consequence is what follows the word “then.”

Note that this does not work the other way unless the statement is biconditional. That is, if the consequence is true, it is impossible to conclude whether the antecedent is true or false.

Similarly, if the antecedent is false, that is not enough information to conclude that the consequence is true.

In mathematical logic, this fact is:

If “$P \rightarrow Q$ is true and $P$ is true, then $Q$ is true.

## Law of Detachment Examples

There are endless examples both in mathematics and beyond of the law of detachment.

One example is a coffee shop that gives a free drink to every 25th customer. As a conditional statement, this is “If someone is the 25th customer, then they get a free drink.”

Thus, if you are the 25th customer, you know you will get a free drink. Likewise, if your friend is the 25th customer, you know he will get a free drink.

Someone could get a free drink in another way. For example, they could have a coupon. Therefore, knowing someone got a free drink is not enough to conclude that they were the 25th customer. Likewise, if someone is not the 25th customer, that is not enough information to know that they did not get a free drink.

## Examples

This section covers common examples of problems involving the law of detachment and their step-by-step solutions.

### Example 1

Suppose the following statement is true:

For every convex quadrilateral, the interior angles add up to $360^{\circ}$.

Which of the following figures can you conclude that the interior angles total $360^{\circ}$?

### Solution

If a figure satisfies the antecedent, it must also satisfy the consequence.

The first two figures, A and B, are not quadrilaterals. Therefore, it is not possible to conclude that the sum of the interior angles to $360^{\circ}$ from the statement.

Figures C, D, and E are all quadrilaterals. Figure D, however, is not convex. Therefore, there is not enough information to make a conclusion.

Since figures C and E are both convex quadrilaterals, the law of detachment says the consequence of the conditional statement is true. Therefore, the total of their interior angles is $360^{\circ}$.

### Example 2

Suppose the following statement is true:

If it rains, then I will bring an umbrella.

Now consider separately that the following are also true. What can be concluded?

A. It is raining.

B. I will bring an umbrella.

### Solution

Consider situation A first.

Since the antecedent of the true conditional statement is true, the consequence must also be true. Therefore, these two statements are enough to conclude that I will bring an umbrella.

In the second case, the statements “if it rains, then I will bring an umbrella” and “I will bring an umbrella” are both true.

In this case, the statement and its consequence are true. Unfortunately, this is not enough information to conclude that it is raining. In fact, no conclusions can be drawn from this information alone.

### Example 3

Suppose the following statement is true:

“It is a gizmo if and only if it is a widget.”

Now consider separately that the following are also true. What can be concluded?

A. It is a gizmo.

B. It is a widget.

### Solution

Note that in this case, the conditional statement is a biconditional statement. Recall that a biconditional statement is one for which $P \rightarrow Q$ and $Q \rightarrow P$ are both true.

Now, first, assume A.

In this case, the biconditional statement being true means that the statement “if it is a gizmo, then it is a widget” is true. Since “it is a gizmo” is the antecedent, and it is true, it is possible to conclude the consequence. Therefore, conclude that it is a widget.

Now, consider case B. Since the biconditional statement is true, the statement “if it is a widget, then it is a gizmo” is also true.

Since the antecedent of this true statement is true, the consequence must also be true. Therefore, it is possible to conclude that it is a gizmo.

### Example 4

Suppose that the following two statements are both true.

- “If it is a cat, then it is a feline.”
- “If it is a feline, then it is a mammal.”

Then, suppose separately that each of the following are true.

A. It is a mammal.

B. The animal is a feline.

C. It is a cat.

### Solution

First, for A, assume that the three statements, “if it is a cat, then it is a feline,” “if it is a feline, then it is a mammal,” and “it is a mammal,” are all true. In this case, “it is a mammal” is the consequence of the second statement. Since it is not an antecedent, there is nothing to conclude.

For B, assume the statements “if it is a cat, then it is a feline,” “if it is a feline, then it is a mammal,” and “it is a feline” are true.

In this case, the first statement is extraneous. “It is a feline” is the antecedent of the second conditional statement. Since it is true and so is the antecedent, the consequence must also be true. Therefore, it is a mammal.

Finally, assume “if it is a cat, then it is a feline,” “if it is a feline, then it is a mammal,” and “it is a cat” are all true. Combining the first statement, “if it is a cat, then it is a feline,” with “it is a cat” is enough to conclude that “it is a feline is true.”

That is not all, however. In this case, it is also important to remember the law of syllogism. This states that if $P \rightarrow Q$ and $Q \rightarrow R$ are both true, then $P \rightarrow R$. Here, since “if it is a cat, then it is a feline” and “if it is a feline, then it is a mammal” are both true, “if it is a cat, then it is a mammal must be true.”

Therefore, since “it is a cat” is true, the consequence “it is a mammal” must also be true.

### Example 5

Use the law of syllogism and the law of detachment to draw conclusions if all of the following are true:

1. “If Dante fails his test, he will fail his class.”

2. “If Adventures in Space is on television, Dante will stay up to watch television.”

3. “Adventures in Space is on television.”

4. “If Dante fails his class, he will get in trouble with his parents.”

5. “If Dante stays up to watch television, then he will fail his test.”

### Solution

The first step, in this case, is to put the statements in a more logical order so that it is easier to apply the law of syllogism.

Statement 3 is the only one that is not a conditional statement, so it should go at the end.

Statement 2 states that “if Adventures in Space is on television, Dante will stay up to watch television.” Since “Adventures in Space is on television” is not the consequence of any other statement, this one is likely first.

In fact, starting at statement 2, it is possible to make a string of statements where the consequence of one is the antecedent of the next.

Statement 2 $\rightarrow$ statement 5 $\rightarrow$ statement 1 $\rightarrow$ statement 4. Then, the end is statement 3.

Using the law of syllogism with the first four statements yields “if Adventures in Space is on television, he (Dante) will get in trouble with his parents.” Since each of the intermediary statements is true, this statement is true.

The antecedent of this statement, “Adventures in Space is on television,” is also true. Therefore, by the law of detachment, the conclusion of the statement, “Dante will get in trouble with his parents,” must also be true.

### Practice Problems

1. Suppose the following statement is true:

“If a prime number is greater than two, then it is odd.”

Which of the following are odd based on this statement?

A. 2

B. 3

C. 5

D. 9

E. 10

2. Suppose the following is a true statement:

“If it is October, then it is fall.”

What can be concluded from the following?

A. It is October 8

B. The date is September 29

C. It is fall

3. Let these statements be true:

“If it is a carrot, it is a vegetable.”

“It is a cabbage.”

What can be concluded?

4. Suppose these statements are true:

“$P \rightarrow Q$.”

“$\neg Q$.”

What can be concluded?

5. Suppose these statements are true:

“If it is cheese, it contains dairy.”

“If it contains dairy, Sonja cannot eat it.”

“It is cheese.”

What can be concluded?

### Answer Key

- Only the numbers 3 and 5 because the number 9 is odd, but the statement doesn’t give enough information to conclude that.
- A means it is fall. Nothing can be concluded from B and C.
- Nothing can be concluded.
- By contrapositive, concluded $\neg P$.
- By the law of detachment, it is dairy. Because of this and the law of syllogism, Sonja cannot eat it.

Images/mathematical drawings are created with GeoGebra.