Statements are tools used in logical reasoning and problem-solving. Compound statement, for instance, not only unites simple statements into one big complex but also still maintains its meaning. This article will examine the compound statements: what they are, their types, and the truth tables for each type.
Table of Content
What is a Statement in Mathematics?
In mathematics, a statement is a sentence that can either be true or false, but it cannot be true and untrue at the same time. It can serve as a logical tool and can be used in the analytical domain, in the creation of assertions on even the abstract and abstract ideas. The statements can either be simple or compound.
For simple statements, one idea is conveyed in a sentence and when they make such a declaration or a statement, it is considered as simple. However, for a compound sentence more than one simple statement is combined using conjunctions like "and," "or," "not," "if-then," and "if and only if".
What is a Compound Statement?
A compound statement is a list of two or more simple statements, each of which is also called an atomic statement, and these simple statements are joined together with logical connectors. The linking words express the degree of association between those statements and the final combined truth value of the compound statement.
Compound statements are used to express complex and detailed ideas which are hard to express. By making simple statements which can represent a diversity of logical relationships possible, compound statements help understand more advanced concepts.
Types of Compound Statements
In this section, we will describe several types of compound statements each of which has its own set of features and logical relation between them. The various types of statements are negation, disjunction, conjunction, conditional, and biconditional.
Negation of a Statement
Negation implies a new statement which is true when the initial statement is false and conversely, when the original statement is true the new statement is false. To put it simply, the negation of a sentence is like translating its truth value into its opposite form. The negation symbol used for statements is "¬" or "~."
Example: Let p be the statement "It is raining." The negation of p, denoted as ¬p or ~p, would be "It is not raining."
Disjunction Statement
A disjunction statement is also known as logic connective "or". The disjunction statement is a compound statement that is true if at least one of the member statements is true. The disjunction symbol used for statements is "∨."
Example: Let p be the statement "It is sunny" and q be the statement "It is raining." The disjunction statement p ∨ q would be "It is sunny or it is raining" (or both).
Conjunction Statement
A conjunction statement, otherwise known as an "and" statement, is a compound condition that is true when both separate statements in the condition are true. The conjunction symbol used for statements is "∧."
Example: Let p be the statement "It is sunny" and q be the statement "It is warm." The conjunction statement p ∧ q would be "It is sunny and it is warm."
Conditional Statement
Conditional sentence, alternatively known as the "if and then” sentence, is a practical sentence which is used to establish a special relation between two statements. This states that if the first statement (the antecedent) is true then the second statement (the consequent) must be true as well. The conditional symbol used for statements is "→."
Example: Let p be the statement "It is raining" and q be the statement "The grass is wet." The conditional statement p → q would be "If it is raining, then the grass is wet."
Bi Conditional Statement
Biconditional statement is a combined statement that is correct when the 'if-then' and the 'then-if' statements are true. In fact, it claims that two sentences or statements are considered logically equivalent. The biconditional symbol used for statements is "↔."
Example: Let p be the statement "It is snowing" and q be the statement "It is cold." The biconditional statement p ↔ q would be "It is snowing if and only if it is cold."
Truth Tables of Compound Statements
Truth tables are a specific way of finding out if the final outcome of the compound statements is true by the truth values of their individual elements. They in fact reflect the possible combinations of the truth values and the final truth value of the compound statement.
Disjunction Truth Table
Disjunction truth table shows the truth values of the disjunction statement (p ∨ q) based on the truth values of p and q.
p | q | p v q |
|---|---|---|
T | T | T |
T | F | T |
F | T | T |
F | F | F |
Conjunction Truth Table
Conjunction truth table shows the truth values of the conjunction statement (p ∧ q) based on the truth values of p and q.
p | q | p ∧ q |
|---|---|---|
T | T | T |
T | F | F |
F | T | F |
F | F | F |
Conditional Truth Table
Conditional truth table shows the truth values of the conditional statement (p → q) based on the truth values of p and q.
p | q | p → q |
|---|---|---|
T | T | T |
T | F | F |
F | T | T |
F | F | T |
Biconditional Truth Table
Biconditional truth table shows the truth values of the biconditional statement (p ↔ q) based on the truth values of p and q.
p | q | p ↔ q |
|---|---|---|
T | T | T |
T | F | F |
F | T | F |
F | F | T |
Conclusion
Compound statements are a fundamental concept in mathematical logic and reasoning. When we apply connectives to the simple statements, the compound statements give more possibilities to develop complex and deeper expressions. Understanding different types of compound statements like negation, disjunction, conjunction, conditional and biconditional will help primarily in the analysis and logical manipulation of the relationships.
Truth tables are a way of systematically finding out the truth value of compound statements and from the truth value of their individual components, thus making them a very useful tool in solving logic problems and helping the thinking process.
Examples of Compound Statements
Example 1: Let p be the statement "It is raining" and q be the statement "The grass is wet." Determine the truth value of the following compound statements:
a) ¬p
b) p ∨ q
c) p ∧ q
d) p → q
e) p ↔ q
Solution:
a) ¬p: The negation of "It is raining" is "It is not raining." If p is false (it is not raining), then ¬p is true.
b) p ∨ q: The disjunction of "It is raining" and "The grass is wet" is "It is raining or the grass is wet (or both)." If at least one of the statements is true, then p ∨ q is true.
c) p ∧ q: The conjunction of "It is raining" and "The grass is wet" is "It is raining and the grass is wet." Both statements must be true for p ∧ q to be true.
d) p → q: The conditional statement "If it is raining, then the grass is wet" is true unless it is raining and the grass is not wet.
e) p ↔ q: The biconditional statement "It is raining if and only if the grass is wet" is true only when both statements have the same truth value (both true or both false).
Example 2: Let p be the statement "The number is even" and q be the statement "The number is divisible by 3." Determine the truth value of the following compound statements for the number 6.
a) ¬p
b) p ∨ q
c) p ∧ q
d) p → q
e) p ↔ q
Solution:
For number 6:
p is true (6 is an even number)
q is false (6 is not divisible by 3)
a) ¬p: The negation of "The number is even" is "The number is not even," which is false for 6.
b) p ∨ q: The disjunction of "The number is even" and "The number is divisible by 3" is true for 6 since 6 is an even number.
c) p ∧ q: The conjunction of "The number is even" and "The number is divisible by 3" is false for 6 since 6 is not divisible by 3.
d) p → q: The conditional statement "If the number is even, then the number is divisible by 3" is false for 6 since 6 is an even number but not divisible by 3.
e) p ↔ q: The biconditional statement "The number is even if and only if the number is divisible by 3" is false for 6 since the two statements have different truth values.