In mathematics, statements are some declarative sentences which can be either true or false but can not be both or neither of them. For example, “I am reading this book”; we cannot decide whether it is true or false; hence it is not a statement. Also, sentences like “Are you going to school?” is not a statement as it is a question.
Some examples of statements:
- 49 is not a perfect square number.
- Sun rises from the east.
- Every even number is a multiple of 2.
- The sum of two odd numbers is again an odd number.
For all the above statements, we decide whether it is true or false. These are examples of simple statements.
When we combine two simple statements using logical connectives, those statements are called compound statements.
For example, let
P: Every even number is a multiple of 2.
Q: 49 is the square of 7.
The simple statements can be combined to form a compound statement using logical connective ‘and’, “Every even number is a multiple of 2 and 49 is the square of 7”.
In Boolean algebra, we have the following logical connectives:
- Conjunction: It is denoted by P Λ Q and read as “P and Q”, where P and Q are simple statements.
- Disjunction: It is denoted by P v Q and read as “P or Q”, where P and Q are simple statements.
- Implication: It is denoted by P → Q and read as “if P then Q”.
- Biconditional: It is denoted by P ↔ Q and read as “P if and only if Q”.
- Negation: It is denoted by ~P or ㄱP and read as “not P”.
Implication statements, also known as conditional statements, follow an “if-then” type of condition to decide whether the compound statement is true or false.
Conditional statements have two parts, the first statement, P, the hypothesis (or antecedent) and the second statement, Q, which is the conclusion (or consequent). It is denoted by “P → Q” or “P ⇒ Q”. For example, “If the number is divisible by 2, then the number is an even number.”
Truth conditions of conditional statements:
|Both hypothesis and conclusion are true.||Implication is true|
|Hypothesis is true, but the conclusion is false||Implication is false|
|Hypothesis is false, but the conclusion is true||Implication is true|
|Both the hypothesis and conclusion are false||Implication is true|
Learn more about conditional statement with examples here.
Tautologies and Contradiction
If a compound statement is true for every possible value, then that statement is known as a tautology. In contrast, if a compound statement is false for every possible value, it is a contradiction.
For example, let us take a compound statement (P → Q) V (Q → P). The truth table for this compound statement is given by:
|P||Q||P → Q||Q → P||(P → Q) v ~ P|
We see that for every truth value, the given statement is true. Hence, the given statement is a tautology.
Some tautology statements:
- For every real number x, (x + √3) and (x – √3) is an irrational number.
- Every whole number is either an even number or an odd number.
- Either he will pass the exams, or he will not pass the exams.
Learn about tautology in maths.
If we take the statement ~(P → Q) Λ ~P, the truth values for this will be:
|P||Q||P → Q||~(P → Q)||~P||~(P → Q) Λ ~ P|
Clearly, the given statement is a fallacy.
Some contradictory statements:
- If x > 1 then x + 2 > 1.
- For a ≠ b, if a divides b then b divides a.