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Logical Operations
In Mathematics, a statement is generally an assertive sentence that is put forward with mainly two direct conclusions. It can either be true or false. Logical operators are used to define the relationship between these statements. A logical operator combines two statements to give rise to a new statement. Examples of Logical operators are: “AND”, “OR”, “NOT”, “IF…THEN”, etc.
These statements used in Mathematics are typically factual. For example, statements like;
1. The sun rises in the east.
2. The sun sets in the west.
These are factual statements and can either have true or false answers. But sentences like;
1. Riya is a good girl.
This is a subjective statement and thus cannot be answered as true or false.
Logical Operators are used for establishing a relationship between such factual sentences. Read further as we delve deeper into logical operators and provide you with intensive study material notes on Statements and Logical Operators.
Types of Logical Operators
There are mainly 5 types of Logical Operators. They are:
1. AND
2. OR
3. NOT
4. IF…THEN
5. IF AND ONLY IF (IFF)
AND Operator
This logical operator is used to join two or more simple statements to form a compound statement. The joining of two statements using the “AND” operator is also known as a conjunction.
The symbol that is used to denote the “AND” operator is ^
Thus, if a and b are two simple statements, then the conjunction of these statements will be denoted symbolically as “a^b” and it will be read as “a AND b”.
Example of AND Logical Operator
Suppose we have two statements;
a = Sun rises in the east
b = 4 is an even number
Then a^b = sun rises in the east AND 4 is an even number.
Traditionally in English grammar, we use the word AND to connect two related sentences, but that need not be the case in logical mathematics. Two unrelated statements can also be combined using the AND operator, as seen in the above example.
OR Operator
Another operator among the five logical operators is the “OR” operator. The OR operator is also used to combine two simple statements. Combining two statements using the OR logical operator is known as a Disjunction or Alternation.
The OR operator is denoted by the symbol “∨”
So if two statements a and b are disjunct, then they are symbolically represented as a ∨ b
And this will be read as: a OR b
Example of OR Operator
Suppose if we have two simple statements,
a = 5 is an odd number
b = 2 is an even number
Now if we combine these two statements using OR operator, it will be
a ∨ b = 5 is an odd number OR 2 is an even number.
NOT Operator
The NOT Operator is used to negate or deny a statement. It basically means the opposite of a statement. For example, If I say “Today is Monday”, then the negation or opposite of the statement shall be “Today is NOT Monday”.
Thus, negation can occur if we place the “NOT” operator at a suitable place in the statement or if we begin a sentence with “it is false that…” or “it is not the case that….”.
The symbol “~” denotes it. So, for a simple statement a, its negation can be written as ~a
Example of NOT operator
Suppose if we have the statement
a = Paris is the capital of France
Then, ~a = Paris is NOT the capital of France
Alternatively, it can also be written as ~a = It is false that Paris is the capital of France.
Although NOT is one of the connective logical operators, it does not connect but rather modifies the statements.
IF…THEN Operator
This is known as a conditional operator. If two simple sentences are combined with “IF..THEN”, it gives a compound statement known as a conditional or implicational statement.
Suppose we have two simple sentences a and b, then “IF…THEN” Operator will be denoted by “a → b” or “a ⇒b”. These symbols are read as “a implies b”, where a is the antecedent and b is the consequent.[2]
Example of If…Then operator
a = PQRD is a square
b = PQ=QR=RD=DP
a ⇒b = If PQRD is a square then PQ=QR=RD=DP
Converse, inverse and contrapositive of a conditional statement
If a and b are two statements, then
Implication or conditional = a ⇒b
Converse = b ⇒ a
Inverse = ~a ⇒ ~b
Contrapositive = ~b ⇒ ~a
Biconditional or Equivalence Statement
The last among the five logical operators, the “IF AND ONLY IF (IFF)” Operator, is used in combining two statements to give a biconditional statement.
Suppose a and b are two statements, then a biconditional statement is denoted by “a ⇔ b” or “a b”
Example of Biconditional or Equivalence Statement
a = a number is divisible by 5
b = the sum of the digits forming the number is also divisible by 5
a ⇔ b = A number is divisible by 5 if and only if the sum of the digits forming the number is also divisible by 5.
Truth Table
As stated before, a statement in mathematics often only has two outcomes: it can either be True or False. These values are known as Boolean values and are used in Boolean Algebra. A truth table encapsulates the different input true and false components of a statement and has a Boolean output accordingly.
It is denoted by the symbols T and F. It is also denoted by the symbols 1 (for the truth) and 0 (for false).
Let us see the truth tables for the unary and binary logical operators.
For unary logical operator: NOT
Since the NOT operator uses a single input, it is known as a Unary logical operator. As we know, the NOT operator returns the opposite value of the statement. So if we enter a statement A as True (T), then the ~A will be False (F) and vice versa. The truth table for the NOT Operator is as follows:
INPUT ( A ) | OUTPUT (~A ) |
T |
F |
F |
T |
Truth Table for Binary Operations
Operators like “AND”, “OR”, “IF…THEN” and “IF AND ONLY IF” have two statements that they combine; thus, they have two inputs. Hence, they are known as binary logical operators.
Truth Table for AND
The conjunction truth table is formed when two statements a and b are combined using the AND Operator. The conjunction value is denoted by a^b, and it can only be true if both the inputs are true.
The truth table for AND is given as:
a | b | a^b |
T | T |
T |
T |
F |
F |
F |
T |
F |
F | F | F |
Thus, in an AND truth table, the output can be true only if both the inputs are true.
Truth Table for OR
The disjunction truth table is created when two statements a and b are combined using the OR Operator. The disjunction value is denoted by a ∨ b.
The truth table for OR is given as
a | b | a ∨ b |
T | T | T |
T | F | T |
F | T | T |
F | F | F |
Thus, in an OR truth table, the output can be false only if both the inputs are false. Even if only one input is true, then it will be true.
Truth Table for Conditional Statements: IF…THEN
The conditional statements are denoted by a ⇒ b
The truth table for conditional statements is as follows:
a | b | a ⇒ b |
T | T | T |
T | F | F |
F | T | T |
F | F | T |
Thus, for a truth table for a conditional statement, the output only depends on the consequent (that is b). If b is true, then the output is true and vice versa. The output is also true if both the inputs are false.
Truth Table for Biconditional Statements: IF AND ONLY IF (IFF)
The biconditional statements are denoted by a ⇔ b
Biconditional statements are formed by the conjunction of two conditional statements, where one conditional statement is the converse of the other. Thus, if we have two statements, a and b, then,
a ⇔ b = (a ⇒ b ) ^ (b ⇒ a)
The truth table for biconditional statements is as follows:
a |
b |
a ⇒ b |
b ⇒ a |
a ⇔ b |
T |
T |
T |
T |
T |
T |
F |
F |
T |
F |
F |
T |
T |
F |
F |
F |
F |
T |
T |
T |
Conclusion
The key takeaway from this study material notes on statements and logical operations is that Logical expressions are mostly used for combining two statements.
There are five types of logical operators. These logical operators can be categorised into different truth tables based on their inputs. The output of each truth table is dependent on the input values.
These statements and logical operations study material aims to provide a comprehensive guide on the topic.
