Logical reasoning finds its applications in several problem-solving strategies in mathematics. It is easy to deduce the conclusions of certain problems on the basis of the facts and through the application of the mathematical principles. In mathematics for solving problems, different logical connectors are used to connect two simple mathematical and logical statements to form compound statements. The two types of connectors used are known as conjunction and disjunction. Conjunctions (“and”) are represented by the mathematical symbol “^ “and disjunctions (“or”) are represented by the mathematical symbol “˅.”
Further, we will discuss different aspects of conjunctions and their applications.
In mathematics, a conjunction refers to a connector added between two statements. The connection is done through the keyword “AND”. The mathematical symbol or the conjunction symbol which represents conjunction is “^”, and this symbol can be read as “AND”.
If we denote two statements as p and q then according to the conjunction meaning, they can be connected by the symbol “^”.
So, it becomes, p ^ q. This compound statement can be read as “p and q”. This statement will be true only if both the statements p and q are true; otherwise, this statement will be false. We will further see all the combinations and conjunction rules by understanding the conjunction truth table.
Let us define a conjunction with an example. For instance,
If our statement 1 is: Karan likes chocolate ice-cream, and our statement 2 is: Riya likes blueberry ice cream, then to connect them we use the connector of conjunction through the keyword “and”. After connecting, our statement becomes, “Karan likes chocolate ice-cream, and Riya likes blueberry ice-cream”. For this statement to be true, both statement 1 and statement 2 need to be true; otherwise, the new statement becomes false.
The statement after adding the conjunction connector “and” will be true only if the individual statements are true in the first place; otherwise, the new statement formed will be false.
The rules are in line with the rules of the AND logic gate.
The symbol for conjunction is “^” which represents the word “AND” which is a type of a logical connector.
When considering statements, we denote them using alphabetical letters when representing them. In that terms, we can define conjunction as, let two statements be p and q. the statement after adding a conjunction connector becomes a compound statement and is represented as “p ^ q”, and it is read as “p and q”.
The truth table is especially important to understand the final values of the compound statements depending on the values of individual statements. All possible combinations are covered in this conjunction truth table. Here “T” letter is used to indicate True value and “F” letter is used to indicate false value.
From the truth table we can clearly deduce the value of the compound statement “p ^ q” will ONLY be true if both, statement p and statement q have true values individually. In all other cases the value of “p ^ q” will be false.
Here are some solved problems for a better understanding of the conjunction meaning and examples.
Let 4 be a rational number and let 7 be a prime number. Is this a conjunction?
Let statement p be that 4 is a rational number
Statement p is TRUE.
Let statement q be that 7 is a prime number
Statement q is TRUE
As per the truth table, if p is True and if q is also true, then “p ^ q” is True
So, in our case, the conjunction “p ^ q” that is “4 is a rational number, and 7 is a prime number” is True.
A: The sun rises in the east
B: It will definitely rain day after tomorrow
Is this a true conjunction?
Statement A which states that the sun rises in the east is a True fact and hence can never be changed. So, statement A is True.
Statement B has the possibility to be false or True. A prediction can never be made with 100% surety that it will definitely rain the day after tomorrow. Thus, statement B has both possibilities. But, for sure, it cannot be proved as a totally True statement at present. Hence, statement B is False.
So, according to the truth table, the conjunction A^B is False.
1. What are the different types of logical connectors in mathematics?
Answer) There are several types of logical connectives used in mathematics for solving problems related to logical reasoning. The commonly used connectives are as follows: -
Negation represented as ~
Equivalence represented as =
Conjunction represented as ^
Disjunction represented as ˅
Implication represented as ->
These are used for important deductions in various fields of engineering, science and mathematics. Their main usage is to form compound statements by joining two or more statements.
2. What is the difference between conjunction and disjunction?
For conjunction, as we have mentioned in the definition of conjunction with the example above, it is a logical connector to connect two statements by using the word “AND”.
While, a disjunction is also a logical connector, but it connects two statements with the keyword “OR”. Here is a truth table for disjunction for a better understanding of the difference between conjunction and disjunction.