LOGIC IN MATHEMATICS WITH REAL EXAMPLES

DEFINITION OF LOGIC IN MATHEMATICS

Logic in mathematics is a very interesting topic. It will enhance your thinking ability in solving mathematics problems. From this article, you will learn some simple logical statements, the five operators of logic, and the truth table of all these operators. Let’s start with the definition of logic and compare it with logic in mathematics.

Logic is the science of thinking about or explaining the reason for something. Logic affords one the ability to express ideas clearly and concisely and to view arguments intelligently and critically. On the other hand, mathematics logic can be defined as the study of the relationship between certain ideal objects such as numbers, functions, geometrical figures, etc. The next paragraph will explain what simple logical statements are and how to identify them.

SIMPLE LOGICAL STATEMENTS

Statements are verbal or written declarations or assertions. The logical property of a statement is that it is either true or false, but not both. So a logical statement is a statement that is either reasonably true or false but not both.

There are logical statements and non-logical statements. The following are examples of logical statements:

Ghana is in Africa;
Abuja is the capital of Nigeria;
6 + 3 =10;
$2\geq 9$ etc.

The above list can be compared with the following which is non-logical statements:

What is your name?
Where is he?
What a great day!

From the above examples, we have seen that questions, exclamations, commands, and expressions of feelings that cannot be either true or false are not logical statements. Since every logical statement is either true or false, a true statement is said to have a truth value true, abbreviated as “T”, while a false statement has a truth value false, abbreviated as “F”.

We usually use the letters $p$ , $q$ , and, $r$ to denote logical statements. For example,

$p$ : the USA is in Asia;
$q$ : ghana is in Africa;
$r$ : 6 + 2 = 8

Here $q$ and $r$ are true statements, therefore have truth values, T. While $p$ is a false statement and has a truth value, F. So for every logical statement, there are two possible truth values, T or F.

FIVE LOGIC OPERATORS IN MATHEMATICS

The following are five main operators of logic in mathematics:

Negation;
Conjunction;
Disjunction;
Implications (or Conditional Statements) and
Bi – Implications (Bi – Conditional Statements)

From the list of operators above, two to five are called compound statements. They can be combined by words like ‘or, and, if and, then, etc. to form a combined statement. In the next few paragraphs, we will discuss the various logic operators with examples and truth tables.

NEGATION OF LOGICAL STATEMENT ( $\sim$ )

The negation of a logical statement ‘ $p$ ‘ is the statement “Not $p$ ” denoted symbolically by “ $\sim$ p”. So if a statement $p$ is true, then $\sim$ $p$ is false and if $p$ is false, $\sim p$ is true. for example, If $p$ is the statement “Ghana is in Africa” then $\sim p$ is the statement “Ghana is not in Africa”.

So if a statement $p$ is true, then $\sim p$ is false and if $p$ is false, $\sim p$ is true. The truth table for $p$ and $\sim p$ is shown below:

The next logic operator we want to discuss is the first among the compound logical statement operator. It is the conjunction.

CONJUNCTION OF LOGICAL STATEMENTS (∧, AND)

Any two simple statements $p$ , $q$ , can be combined by the word “and” to form a compound statement “ $p$ and $q$ ” called the conjunction of $p$ , $q$ and it is denoted symbolically by $p$ ∧ $q$ . For example, let $p$ be ” The day is bright” and $q$ be ” so the sun will rise” then $p$ ∧ $q$ is the statement “the day is bright and so the sun will rise”.

The symbol “∧” can be used to define the intersection of two sets A and B as follows : A ∩ B= { $x$ : $x$ ∈ A ∧ $x$ ∈ B}

The truth table value of $p$ ∧ $q$ is shown below:

In summary, from the table above, $p$ ∧ $q$ is only True if statement $p$ and statement $q$ is true.

We are going to consider the disjunction of logical statement which is also part of compound logical statements .

DISJUNCTION OF LOGICAL STATEMENTS (∨, OR)

Any two simple statements $p$ , $q$ can be combined by the word “or” to form a new statement “ $p$ or $q$ ” called disjunction of $p$ , $q$ and written symbolically as $p$ ∨ $q$ . For example, let $p$ be “Ade studied English” and $q$ be “James studied Biology” then $p$ ∨ $q$ is is the statement “Ade studied English or James studied Biology”.

The symbol can also be used to determine the Union of two sets A and B: A ∪ B= { $x$ : $x$ ∈ A ∨ $x$ ∈ B}

Using the same logical statements used above, the truth table disjunction of logical statements ( $p$ ∨ $q$ ) is:

In Summary, from the truth table, $p$ ∨ $q$ is only FALSE if $p$ and $q$ are false, otherwise it will be TRUE.

The next, we will look at the conditional statement (Implications). This is also part of compound logical statements.

CONDITIONAL STATEMENT (IMPLICATIONS) (IF, THEN, →)

Are logical statements with antecedent and consequence. it is denoted by $p$ $\rightarrow q$ . The “If” statement (i.e $p$ ) is called the antecedent, while the “then” statement (i.e. $q$ ) is called the consequent. For example, let $p$ be “She leaves early and $q$ ‘She will meet the prince” then $p\rightarrow q$ is the statement “if She leaves early, then she will meet the Prince”.

The truth table for conditional statement is as shown below:

conditional logical statement (implication)

In summary, from the table above, $p\rightarrow q$ is only FALSE, if $p$ is true and $q$ false.

We will consider other aspect of conditional statements, which are the converse of a conditional statement, inverse of a conditional statement and the contrapositive of a conditional statement.

Converse of a Conditional Statement: Given the conditional statement $p\rightarrow q$ , then the converse of $p\rightarrow q$ is the statement $q\rightarrow p$ . For example, let $p$ be “Bola is a good girl” and $q$ be $2+2=4$ , then $p\rightarrow q$ will be the statement “If Bola is a girl then $2+2=4$ “. The Converse $q\rightarrow p$ is a statement “If $2=2=4$ then Bola is a girl”.
Inverse of a Conditional Statement: The inverse of $p\rightarrow q$ is the statement $\sim p\rightarrow \sim q$ . Using the $p$ and $q$ statements above as example, the inverse of $p\rightarrow q$ is the statement “If Bola is not a girl then $2+2\neq 4$ .
Contrapostive of a Conditional Statement: The contrapositive statement of $p\rightarrow q$ is $\sim q\rightarrow \sim p$ (i.e. the converse of the inverse). Also using the statement of $p$ and $q$ above, the contrapositive of $p\rightarrow q$ is the statement “If $2+2\neq 4$ then Bola is not a girl”.

The summary of all aspect of conditional statements in shown in the table below:

BI – CONDITIONAL STATEMENTS (EQUIVALENCE) (IF AND ONLY IF, $\leftrightarrow$ )

They are statements of the form “ $p$ if and only if”. It is the combination of two conditional statements and so it is bi-conditional or equivalence statement. It is denoted by the symbol, $p\leftrightarrow q$ . For example, let $p$ be “America is a rich Country” and $q$ be “ $2+3=5$ “, then $p\leftrightarrow q$ is the statement “America is a rich Country if and only if “ $2+3=5$ “.

The truth table for bi-conditional statements is as shown below:

In summary, from the table above, $p\leftrightarrow q$ is TRUE, if $p$ and $q$ are both true or both false. Otherwise it is FALSE.

From our discussion so far, we have seen that the simple and concise way of determining the real values of both negation statements and all compound statements is usually by constructing the truth table of the statements. Let us consider some examples on this topic. Some of these examples you will most likely see in your examinations if you are sitting for WASSCE or GCE conducted in Nigeria and some West Africa countries. Desire to get past questions and answers? Get JAMB past question and answer with keynotes,

LOGIC IN MATHEMATICS EXAMPLES

Let us consider some examples on this topic. We will look at two video examples and one practical example. After considering this examples, try doing the quizzes in the worksheet. Click HERE to download the worksheet.

Sample 1: Construct the truth table of ( $p\rightarrow q$ ) ∧ ( $q\rightarrow p$ ).

Before we start constructing the truth table, let us first understand the logical symbols we are asked to work with in this question.

Firstly we are given the Conjunction of logical statement which is denoted by the logical and “∧” and the conditional or implication of logical statement which is denoted by “→”.

Secondly, lets consider under what condition they can be True or False. In this regard, we should note that:

From the Conditional statement $p\rightarrow q$ is only FALSE if $p$ is true and $q$ is false, otherwise, it will be TRUE;
For Conjunction of logical statement, $p$ ∧ $q$ is only TRUE if $p$ and $q$ is true, otherwise it will be FALSE.

Now with this information, let’s draw the truth table:

conjunction 0f logical statement truth table

Sample 2: Confirm if then Conditional Statement (Implication) $x\rightarrow y$ is equivalent to $\sim y\rightarrow \sim x$ .

Watch the video below for the solution to this question.

Sample 3: Let $p$ be “The weather is cold” and let $q$ be “The sun is shining”. Give a simple verbal statement which describe the logic $\sim p$ ∧ $q$ .

Watch the video below for the solution to this question.

For more questions on logic in mathematics, click HERE to download the logic in mathematics pdf worksheet. Try the quizzes so as to master all you have learnt in this article.

In conclusion, we have learnt that simple logical statements is a statement that is either reasonably “true” or “false” but not both, also denoted by $p$ , $q$ and $r$ , and above all, we have seen that any two simple logical statements, $q$ can form a compound statement and lots more.

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August 5, 2021

MATH ARTICLES

LOGIC IN MATHEMATICS

DEFINITION OF LOGIC IN MATHEMATICS

SIMPLE LOGICAL STATEMENTS

FIVE LOGIC OPERATORS IN MATHEMATICS

NEGATION OF LOGICAL STATEMENT ( $\sim$ )

CONJUNCTION OF LOGICAL STATEMENTS (∧, AND)

DISJUNCTION OF LOGICAL STATEMENTS (∨, OR)

CONDITIONAL STATEMENT (IMPLICATIONS) (IF, THEN, →)

BI – CONDITIONAL STATEMENTS (EQUIVALENCE) (IF AND ONLY IF, $\leftrightarrow$ )

LOGIC IN MATHEMATICS EXAMPLES

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NEGATION OF LOGICAL STATEMENT ( $\sim$ )

BI – CONDITIONAL STATEMENTS (EQUIVALENCE) (IF AND ONLY IF, $\leftrightarrow$ )