Download Validity of a Conditional Statement

Mathematical Logic
A conditional statement is a statement that takes the form of "if p, then q".
If a conditional statement is true, then any time p is true, q must also be true.
Example:
The conditional statement below is true.
If the measure of an angle is less than 90°, then it is acute.
Based on the conditional statement, what can be concluded from the additional statement "m∠X = 67°"?
Answer:
In this conditional statement, p (the hypothesis) is represented by the statement "the measure of an angle is less than
90°", and q (the conclusion) is represented by the statement "it is acute".
For the given additional statement, p is true. Therefore, q must also be true because it is given that the conditional
statement is true.
So, ∠X is acute.
The converse of a conditional statement, "if p, then q", is found by interchanging the hypothesis
and conclusion, "if q, then p."
Example: What is the converse of the statement below?
If the measure of an angle is less than 90°, then it is acute.
Answer:
The converse of a conditional statement is formed by interchanging the hypothesis and the conclusion of the
conditional statement.
The converse of the given conditional statement is shown below.
If an angle is acute, then its measure is less than 90°.
Notice that in this case the converse of the statement is true. However, this is not always the case.
The inverse of a conditional statement, "if p, then q", is found by negating both the hypothesis
and conclusion, "if not p, then not q."
Example: What is the inverse of the statement below?
If the measure of an angle is less than 90°, then it is acute.
Answer:
The inverse of a conditional statement is formed by negating both the hypothesis and the conclusion of the
conditional statement.
The inverse of the given conditional statement is shown below.
If the measure of an angle is not less than 90°, then it is not acute.
Notice that in this case the inverse of the statement is true. However, this is not always the case.
The contrapositive of a conditional statement, "if p, then q", is found by interchanging and
negating both the hypothesis and conclusion, "if not q, then not p."
Example: What is the contrapositive of the statement below?
If the measure of an angle is less than 90°, then it is acute.
Answer:
The contrapositive of a conditional statement is formed by interchanging and negating both the hypothesis and the
conclusion of the conditional statement.
The contrapositive of the given conditional statement is shown below.
If an angle is not acute, then its measure is not less than 90°.
Notice that the contrapositive of the statement is true.
Any time a conditional statement is true, its contrapositive will also be true.
Counterexamples
It only takes one counterexample to refute, or disprove, an assertion.
A counterexample refutes, or disproves, an assertion. Counterexamples are often used in
math, logic, and philosophy.
Example 1:
If x < y, and y = 4, then x ≤ 3.
(For all real numbers x and y)
What is a counterexample that would make the assertion above false?
Solution: x = 3.5
If x is smaller than y, which is 4, then x could be less than or equal to 3. However, numbers 3.1 to 3.9 are also less
than 4, yet they are greater than 3. Therefore x = 3.5 is a counterexample to the assertion.
Example 2:
A student made the conjecture below.
In any quadrilateral, the oppossing angles are congruent.
Determine if the conjecture is true or false. Provide a proof if the statement is true, or provide a counterexample if the
statement is false.
Solution:
The statement is false. In a trapezoid, the opposing angles are not necessarily congruent as shown below.
Validity of a Conditional Statement
A valid argument guarantees that if the premises are all true, then the conclusion is also true.
The form of an argument, not its content or truth value, determines its validity.
A conditional statement is of the following form.
If p, then q.
A valid argument for a conditional statement is of the following forms.
p. Therefore, q.
Not q. Therefore, not p.
An invalid argument for a conditional statement is of the following forms.
q. Therefore, p.
Not p. Therefore, not q.
Example:
If the sum of the interior angles of a polygon equals 540°, then it is a pentagon.
Which of the following arguments is valid with respect to the conditional statement above?
1. A polygon is a pentagon. Therefore, the sum of its interior angles equals 540°.
2. The sum of the interior angles of a polygon equals 540°. Therefore, it is a pentagon.
3. A polygon is not a pentagon. Therefore, the sum of its interior angles does not equal 540°.
4. The sum of the interior angles of a polygon does not equal 540°. Therefore, it is not a pentagon.
Solution:
A valid argument guarantees that if the premises are all true, then the conclusion is also true. The form of an
argument, not its content or truth value, determines its validity.
Determine the forms of the four arguments given.
Argument 1 is of the form "q. Therefore, p."
Argument 2 is of the form "p. Therefore, q."
Argument 3 is of the form "not q. Therefore, not p."
Argument 4 is of the form "not p. Therefore, not q."
Therefore, arguments 2 and 3 are valid.
Deductive Reasoning
Deductive reasoning is a tool used to take a given set of facts and use them to produce a new
fact.
Example: Quadrilateral PQRS is a parallelogram. If adjacent sides are congruent, what must be true?
Solution:
It is given that quadrilateral PQRS is a parallelogram. Thus, opposite angles are congruent and opposite sides are
congruent.

∠PQR ≅ ∠RSP

∠SPQ ≅ ∠QRS

Line Segment PQ ≅ Line Segment RS

Line Segment QR ≅ Line Segment SP
If adjacent sides are also congruent then the following must be true.
Segment PQ ≅ Segment QR ≅ Segment RS ≅ Segment SP
Since all 4 sides are congruent, PQRS is a rhombus.