In mathematics, a statement is a declarative sentence that consists mostly of two direct conclusions. It could be either true or false. The relationship between these statements is defined using logical operators. A logical operator joins two assertions together to form a new one. “AND,” “OR,” “NOT,” “IF…THEN,” and other logical operators are examples.
These propositions in mathematics are almost always true. For instance, consider the following statements:
1. The east is where the sun rises.
2. The west is where the sunsets.
These are true or false statements with true or false replies. However, phrases like;
1. Riya is a lovely young lady.
Because this is a personal remark, it cannot be responded to as true or false.
For establishing a relationship between such factual sentences, logical operators are used. Continue reading to learn more about logical operators and to get detailed study notes on Statements and Logical Operators.
Logical Operators Types
Logical Operators are divided into five categories. They are:
1. AND
2. OR
3. NOT
4. IF…THEN
5. ONLY IN THE EVENT THAT (IFF)
Operator AND
This logical operator joins two or more basic assertions together to generate a compound statement. Conjunction refers to the connecting of two statements using the “AND” operator.
The “AND” operator is represented by the symbol “AND.”
If a and b are two simple statements, the conjunction of these statements will be symbolically indicated as “ab” and interpreted as “a AND b.”
AND Logical Operator Example
Consider the following two statements:
a = The sun rises eastward.
The number b = 4 is an even number.
The sun rises in the east, and 4 is an even number, so ab = sun rises in the east.
We employ the phrase AND to connect two related statements in English grammar, but this isn’t necessary in logical mathematics. The AND operator can also be used to merge two unconnected assertions, as shown in the example above.
Operator OR
The “OR” operator is another one of the five logical operators. When two basic assertions are combined, the OR operator is employed. A Disjunction or Alternation is when two statements are combined using the OR logical operator.
The symbol “” stands for the OR operator.
If two propositions a and b are mutually exclusive, they are symbolically represented as a and b.
And the following will be read: an OR b
OR Operator Example
Consider the following two simple statements:
The number 5 is an odd number.
The number b = 2 is an even number.
When we combine these two statements with the OR operator, we get
a b = 5 is either an odd or an even number.
Operator NOT
A statement is negated or denied using the NOT Operator. It is the polar opposite of a statement. If I say “Today is Monday,” for example, the negation or opposite of that sentence is “Today is NOT Monday.”
Thus, negation can occur if the “NOT” operator is used appropriately in the statement or if a sentence begins with “it is false that…” or “it is not the case that…”.It is indicated by the sign “”. As a result, the negation of the simple assertion a can be represented as a.
NOT operator example
Assume we have the following statement.
a = France’s capital is Paris.
Then a = Paris is NOT France’s capital.
It can also be written as a = The claim that Paris is France’s capital is false.
Although NOT is a connective logical operator, it does not connect assertions; rather, it alters them.
Operator IF…THEN
A conditional operator is what this is called. A conditional or implicational statement is created when two basic statements are coupled with “IF…THEN.”
If we have two simple sentences a and b, then the “IF…THEN” Operator will be represented by “a b” or “ab.” The meaning of these symbols is “a implies b,” with a being the antecedent and b being the consequent.
If…Then operator example
The square a = PQRD
PQ=QR=RD=DP = b
a b = PQ=QR=RD=DP if PQRD is a square.
A conditional statement’s inverse, inverse, and contrapositive
If both a and b are true, then
Conditional or implication = a b
b a = converse
a + b = inverse
b a Contrapositive = b a
Statement of Biconditionality or Equivalence
The “IF AND ONLY IF (IFF)” operator, the last of the five logical operators, is used to combine two assertions into a biconditional statement. If a and b are two statements, then “a b” or “a b” denotes a biconditional statement.
Biconditional or Equivalence Statement Example
a = a number that can be divided by five
b = the number’s sum of digits is also divisible by five.
a b = A number is divisible by five if and only if the sum of its digits is also divisible by five.
NOT is a unary logical operator
The NOT operator is an Unary logical operator because it only takes one argument. The NOT operator, as we all know, returns the statement’s opposite value. As a result, if we input a statement A as True (T), the A will become False (F), and vice versa.
Operators like “AND,” “OR,” “IF…THEN,” and “IF AND ONLY IF” combine two assertions, resulting in two inputs. As a result, they are referred to as binary logical operators.
Table of Contents for AND
When the AND Operator is used to join two statements a and b, the conjunction truth table is created. The conjunction value is represented by the letters ab, and it can only be true if both inputs are true. As a result, the output of an AND truth table can only be true if both inputs are true.
Table of Contents for OR
When two propositions a and b are merged using the OR Operator, the disjunction truth table is formed. A b represents the disjunction value. The output of an OR truth table can only be false if both inputs are false. It will be true even if only one input is true.
IF…THEN Truth Table for Conditional Statements
A b is used to signify conditional sentences. As a result, the output of a truth table for a conditional statement is solely dependent on the consequent (that is b). The output is true if b is true, and vice versa. If both inputs are false, the result is also true.
Biconditional Statements Truth Table: IF AND ONLY IF (IFF)
A b is used to denote biconditional assertions. Biconditional statements are created by combining two conditional statements, one of which is the polar opposite of the other. As a result, if we have two statements, a and b,
a ⇔ b = (a ⇒ b) ^ (b ⇒ a)
Conclusion
The most important takeaway from this study material on statements and logical operations is that logical expression are typically used to combine two statements. Logical operators are divided into five categories. Based on their inputs, these logical operators can be classified into different truth tables. Each truth table’s output is determined by the input values. The purpose of this study material on statements and logical operations is to provide a complete overview of the subject.