Determine a counter-example to negate the truth of a particular statement.

Statements

3.1

Statements

Definition

A statement is a sentence of which the truth value can be determined, that is either true or false, but not both.

Extra Information

Questions, explanation and command sentences are not statements because their truth values cannot be determined.

Example: Determine whether each of the following sentences is a statement or not, and provide the reason.

Question	Answer
\(4 + 3 = 8\)	Statement, it is a false statement.
The pentagon has \(5\) sides.	Statement, it is a true statement.

Example: Determine whether the following mathematics statements are true or false.
If a statement is false, explain why.

Question	Answer
\(3\) is a prime number.	True statement.
\(–11 > –8\)	False statement because \(-8\) is bigger than \(-11\).
\(5\) is a factor to \(8\).	False statement because \(5\) cannot be a factor to \(8\). Factors of \(5\) are numbers which on dividing \(5\) leave no remainder.

Tips

Not all the mathematical statements are true.
The truth values of the mathematical statements can be determined.

Negate a Statement

Explanation

Use the word "no" or "not" to negate a statement.
The negation of statement \(p\) is written as \(\sim p\).

Example: Form a negation (~p) for each of the following statements (p) by using the word "no" or "not".

Question	Answer
\(13\) is a multiple of \(5\).	\(13\) is not a multiple of \(5\).
\(20\) is a prime number.	\(20\) is not a prime number.

Determine the Truth Value of a Compound Statement

Definition

A compound statement is a combination of two or more statements by using the word "and" or "or".
The word "and" in a mathematical statement mean both while the word "or" means one of them or both.

Example: Determine the Truth Value of a Compund Statement
Combine the following statements, \(q\) and \(r\) by using the words "and" or "or".

\(q\): A pentagon has two diagonals. \(r\): A heptagon has four diagonals.
and	or
A pentagon has two diagonals and a heptagon has four diagonals.	A pentagon has two diagonals or a heptagon has four diagonals.

Truth Table

\(p\)	\(q\)	\(p\) and \(q\)	\(p\) or \(q\)
True	True	True	True
True	False	False	True
False	True	False	True
False	False	False	False

Example: Determine the Truth Values of the Compound Statements

Q uestion:

\(x+3\lt x-5\) and \(99\) is an odd number.

Solution:

	Statement	Truth Value
\(p\) \(q\) \(p\) and \(q\)	\(x+3\lt x-5\). \(99\) is an odd number. \(x+3\lt x-5\) and \(99\) is an odd number.	False True False

Question:

The sum of interior angles of a triangle or a quadrilateral is \(360\degree\).

Solution:

	Statement	Truth Value
\(p\) \(q\) \(p\) or \(q\)	The sum of interior angles of a triangle is \(360\degree\). The sum of interior angles of a quadrilateral is \(360\degree\). The sum of interior angles of a triangle or a quadrilateral is \(360\degree\).	False True True

Construct a Statement in the Form of an Implication

Implication "If \(p\), then \(q\)"

A statement "if \(p\), then \(q\)" is known as an implication where

\(p\) is denoted as the antecedent.
\(q\) is denoted as the consequent.

Example:

Determine the antecedent and consequent for the following implications "if \(p\), then \(q\)".

If \(x\) is a factor of \(16\), then \(x\) is a factor of \(64\).

Solution:

Antecedent: \(x\) is a factor of \(16\).

Consequent: \(x\) is a factor of \(64\).

Implication "\(p\) if and only if \(q\)"

An implication "\(p\) if and only if \(q\)" consists of the following two implications:

if \(p\), then \(q\)
if \(q\), then \(p\)

Example:

Write two implications based on the implication "\(p\) if and only if \(q\)" given below.

\(\sqrt{r}=15\) if and only if \(r=225\).

Solution:

Implication 1: If \(\sqrt{r}=15\), then \(r=225\).

Implication 2: If \(r=225\), then \(\sqrt{r}=15\).

Construct and Compare the Truth Value of Converse, Inverse and Contrapositive of an Implication

Statement	If \(p\), then \(q\).
Converse	If \(q\), then \(p\).
Inverse	If \(\sim p\), then \(\sim q\).
Contrapositive	If \(\sim q\), then \(\sim p\).

Example

Write the converse, inverse and contrapositive of the following implication.

If is a positive number, then \(x\) is greater than \(0\).

Solution:

Statement	If \(x\) is a positive number, then \(x\) is greater than \(0\).
Converse	If \(x\) is greater than \(0\), then \(x\) is a positive number.
Inverse	If \(x\) is not a positive number, then \(x\) is not greater than \(0\).
Contrapositive	If \(x\) is not greater than \(0\), then \(x\) is not a positive number.

Tips

\(\sim p\) is a complement of \(p\).
Then, the complement of \(p^2-q^2>0\) is \(p^2-q^2\leq0\).

Determine a Counter-Example to Negate The Truth of a Particular Statement

For a false statement, at least one counter-example can be given to negate the truth of that statement.

Example

Determine the truth value of the following mathematical statements. If it is false, give one counter-example to support your answer.

Question: The sum of interior angles of all polygons is \(180\degree\).

Answer: False because the sum of interior angles of a pentagon is \(540\degree\).

Question: \(6\) or \(36\) is a multiple of \(9\).

Answer: True

Statements

3.1

Statements

Definition

A statement is a sentence of which the truth value can be determined, that is either true or false, but not both.

Extra Information

Questions, explanation and command sentences are not statements because their truth values cannot be determined.

Example: Determine whether each of the following sentences is a statement or not, and provide the reason.

Question	Answer
\(4 + 3 = 8\)	Statement, it is a false statement.
The pentagon has \(5\) sides.	Statement, it is a true statement.

Example: Determine whether the following mathematics statements are true or false.
If a statement is false, explain why.

Question	Answer
\(3\) is a prime number.	True statement.
\(–11 > –8\)	False statement because \(-8\) is bigger than \(-11\).
\(5\) is a factor to \(8\).	False statement because \(5\) cannot be a factor to \(8\). Factors of \(5\) are numbers which on dividing \(5\) leave no remainder.

Tips

Not all the mathematical statements are true.
The truth values of the mathematical statements can be determined.

Negate a Statement

Explanation

Use the word "no" or "not" to negate a statement.
The negation of statement \(p\) is written as \(\sim p\).

Example: Form a negation (~p) for each of the following statements (p) by using the word "no" or "not".

Question	Answer
\(13\) is a multiple of \(5\).	\(13\) is not a multiple of \(5\).
\(20\) is a prime number.	\(20\) is not a prime number.

Determine the Truth Value of a Compound Statement

Definition

A compound statement is a combination of two or more statements by using the word "and" or "or".
The word "and" in a mathematical statement mean both while the word "or" means one of them or both.

Example: Determine the Truth Value of a Compund Statement
Combine the following statements, \(q\) and \(r\) by using the words "and" or "or".

\(q\): A pentagon has two diagonals. \(r\): A heptagon has four diagonals.
and	or
A pentagon has two diagonals and a heptagon has four diagonals.	A pentagon has two diagonals or a heptagon has four diagonals.

Truth Table

\(p\)	\(q\)	\(p\) and \(q\)	\(p\) or \(q\)
True	True	True	True
True	False	False	True
False	True	False	True
False	False	False	False

Example: Determine the Truth Values of the Compound Statements

Q uestion:

\(x+3\lt x-5\) and \(99\) is an odd number.

Solution:

	Statement	Truth Value
\(p\) \(q\) \(p\) and \(q\)	\(x+3\lt x-5\). \(99\) is an odd number. \(x+3\lt x-5\) and \(99\) is an odd number.	False True False

Question:

The sum of interior angles of a triangle or a quadrilateral is \(360\degree\).

Solution:

	Statement	Truth Value
\(p\) \(q\) \(p\) or \(q\)	The sum of interior angles of a triangle is \(360\degree\). The sum of interior angles of a quadrilateral is \(360\degree\). The sum of interior angles of a triangle or a quadrilateral is \(360\degree\).	False True True

Construct a Statement in the Form of an Implication

Implication "If \(p\), then \(q\)"

A statement "if \(p\), then \(q\)" is known as an implication where

\(p\) is denoted as the antecedent.
\(q\) is denoted as the consequent.

Example:

Determine the antecedent and consequent for the following implications "if \(p\), then \(q\)".

If \(x\) is a factor of \(16\), then \(x\) is a factor of \(64\).

Solution:

Antecedent: \(x\) is a factor of \(16\).

Consequent: \(x\) is a factor of \(64\).

Implication "\(p\) if and only if \(q\)"

An implication "\(p\) if and only if \(q\)" consists of the following two implications:

if \(p\), then \(q\)
if \(q\), then \(p\)

Example:

Write two implications based on the implication "\(p\) if and only if \(q\)" given below.

\(\sqrt{r}=15\) if and only if \(r=225\).

Solution:

Implication 1: If \(\sqrt{r}=15\), then \(r=225\).

Implication 2: If \(r=225\), then \(\sqrt{r}=15\).

Construct and Compare the Truth Value of Converse, Inverse and Contrapositive of an Implication

Statement	If \(p\), then \(q\).
Converse	If \(q\), then \(p\).
Inverse	If \(\sim p\), then \(\sim q\).
Contrapositive	If \(\sim q\), then \(\sim p\).

Example

Write the converse, inverse and contrapositive of the following implication.

If is a positive number, then \(x\) is greater than \(0\).

Solution:

Statement	If \(x\) is a positive number, then \(x\) is greater than \(0\).
Converse	If \(x\) is greater than \(0\), then \(x\) is a positive number.
Inverse	If \(x\) is not a positive number, then \(x\) is not greater than \(0\).
Contrapositive	If \(x\) is not greater than \(0\), then \(x\) is not a positive number.

Tips

\(\sim p\) is a complement of \(p\).
Then, the complement of \(p^2-q^2>0\) is \(p^2-q^2\leq0\).

Determine a Counter-Example to Negate The Truth of a Particular Statement

For a false statement, at least one counter-example can be given to negate the truth of that statement.

Example

Determine the truth value of the following mathematical statements. If it is false, give one counter-example to support your answer.

Question: The sum of interior angles of all polygons is \(180\degree\).

Answer: False because the sum of interior angles of a pentagon is \(540\degree\).

Question: \(6\) or \(36\) is a multiple of \(9\).

Answer: True

Statements

Statements

Chapter : Logic Reasoning

Topic : Determine a counter-example to negate the truth of a particular statement.

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