Conjunctions (Math)

• Definition
• Example
• Notation
• Rules for outcomes of conjunctions
• Truth tables

Definition:

A conjunction is a group of two or more statements in mathematics, separated by the logical word 'AND'. These statements can be true or false. The result of combining these statements with the help of the 'AND' operator is a compound statement called a conjunction. The outcome of this compound statement (or conjunction) can also be either true, or false.

Example:

Statement 1: "George plays football" 

Statement 2: "Jeremy plays football" 

Conjunction: Statement 1 ^ Statement 2 (read as "statement 1 and statement 2")

Notation:

Conjunctions are written by using the symbol  '^' which means AND.

Rules for outcomes of conjunctions:

A conjunction is true if and only if all of its statements are themselves true. For instance, in the above example,

• if statement 1 is true and statement 2 is true also, then the conjunction "statement 1 ^ statement 2" is true.
• If either statement 1 is false or statement 2 is false, then the conjunction "statement 1 ^ statement 2" is false.
• If both the statements are false, then the conjunction "statement 1 ^ statement 2" is false.

Truth table:

Truth tables can be constructed in conjunctions. A truth table explores all the different possible combinations of the statements' truth value. For example, if there are two statements, then all the possible combinations of their truth values are "True - True", "True - False", "False - True" and "False - False". If there are three statements, then all the possible combinations of their truth values as follows:

• True, True,True
• True, True, False
• True, False, True
• True, False, False
• False, True, True
• False, True, False
• False, False, True
• False, False, False

For instance, the truth table for the above two statements (in the above example) is as follows:

Statement 1 : "George plays football"

Statement 2 : "Jeremy plays football"

~ Truth Table ~

Statement 1	Statement 2	Statement 1 ^ Statement 2
True	True	True
True	False	False
False	True	False
False	False	False

In the above truth table,

• in the first row, Statement 1 is considered "True" and Statement 2 is considered "True". The result is "True" 

• in the second row, Statement 1 is considered "True" and Statement 2 is considered "False". The result is "False"

• in the third row, Statement 1 is considered "False" and Statement 2 is considered "True". The result is "False" 

• in the fourth row, Statement 1 is considered "False" and Statement 2 is considered "False". The result is "False"