You might have learned about Sentences in your English classes. A sentence is a group of words which arrange themselves in a meaning manner. Any sentence which we say or write or think is a statement. You will be surprised that in mathematics also we have some statements. These are the mathematically valid statements. In this section, we are going to discuss mathematical reasoning and mathematical statement. Let us start to learn more!
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Statements
A mathematical statement is the basic unit of any mathematical reasoning. A mathematical reasoning is either inductive (mathematical induction) or deductive. We will discuss deductive reasoning in this section. Any assertive sentence which is either true or false but not both is a mathematically acceptable statement. This type of statement is a valid statement.
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A statement is always denoted by a small letter like a, p, x, z etc. Any ambiguous sentence is not a statement and hence it is invalid. Consider a sentence t: 40 is a square root of 1600. Since it is true hence, it is a mathematical statement.
Properties of Statements
- A statement is a mathematical acceptable sentence
- Mathematical statements are either true or false, they cannot be both
- Any exclamatory sentences are not mathematical statements
- An imperative sentence is not a valid mathematical statement
- An interrogative sentence is not a mathematical valid statement
- A sentence involving variables such as here, there, today, tomorrow, and yesterday or any pronouns is not a statement.
- Ambiguous sentences are invalid in mathematics
Types of Statements
There are two types of statements:
Open Statements
A sentence which contains one or more variable such that when certain values are given to the variable it becomes a statement is an open statement.
Compound Statements
Two or more simple statements combine to form a compound statement. Each statement in the compound statement is a component statement. The simple statements are combined with the use of words such as ‘and (∧)’, ‘or (∨)’, ‘not (∼)’, ‘if … then (→)’, ‘only if’, ‘if and only if (↔)’.
- The compound statement with the word ‘and’ is true if all its component statements are true
- If any of the component statements of the compound statement with ‘and’ is false then the statement is false.
- A compound statement with ‘or’ is true when one or both the component statement(s) is true.
- When both the component statements are false then the compound statement with ‘or’ is false.
The word ‘OR’
The word ‘or’ can be used in two ways
- The inclusive “OR”: This means either one or both the component statements are included.
- The exclusive “OR”: It implies that only one of the component statement is included and not the both.
Negation of Statements
The denial of the statements is basically called the negation of the statements. It is denoted by the sign (∼). It is read as ‘not’. If p: Number 7 is an odd number. The negation of p is not p, ∼ p: Number 7 is not an odd number.
Truth Tables
I. The truth table for the truth values of a and b (a ∧ b), and b and a (b ∧ a)
| a | b | a ∧ b | b ∧ a |
| T | T | T | T |
| T | F | F | F |
| F | T | F | F |
| F | F | F | F |
where T stands for TRUE and F stands for FALSE. a and b (a ∧ b) is true when a and b are both true.
II. The truth table for the truth values of a or b (a ∨ b), and b or a (b ∨ a)
| a | b | a ∨ b | b ∨ a |
| T | T | T | T |
| T | F | T | T |
| F | T | T | T |
| F | F | F | F |
a or b (a ∨ b) is false when a and b are both false.
III. The truth table for the negation of a statement
| a | ∼a | ∼(∼a) |
| T | F | T |
| F | T | F |
Implications
Implications are the statements with the words ‘if … then’, ‘only if’, and ‘if and only if (iff)’.
The word ‘If… then’
‘If … then’ type of compound statements are the conditional statements. They are denoted by a → b (or, a ⇒ b). This means
- a is sufficient for b
- b is necessary for a
- a leads to b
- a exists only if b
- b exists if a
- if a then b
Truth Table for a → b
| a | b | a → b | b → a |
| T | T | T | T |
| T | F | F | T |
| F | T | T | F |
| F | F | T | T |
The word ‘If and Only If’
‘If and only if’ type of compound statements are the equivalence statement. It is denoted by a ↔ b or a ⇔ b. This means
- a is a necessary and a sufficient condition for b.
- b is a necessary and a sufficient condition for a.
- If a then b and if b then a.
- a if and only if b.
- b if and only if a.
Truth Table for a ↔ b
| a | b | a ↔ b | b ↔ a |
| T | T | T | T |
| T | F | F | F |
| F | T | F | F |
| F | F | T | T |
Contrapositive and Converse
These are the statements which can be formed from a given statement with ‘if … then’.
- The contrapositive of the statements ‘if a, then b’ are given by ‘if ∼b, then ∼a’.
- The converse of the statements ‘if a, then b’ are given as ‘if b, then a’.
The Truth Table for Contrapositive and Converse
| a | b | ∼a → ∼b | b ↔ a |
| T | T | T | T |
| T | F | F | T |
| F | T | T | F |
| F | F | T | T |
Validating Statements
There are some rules for checking if the statements are true or not.
Rule 1. Statements with ‘AND’
If a and b are mathematical statements, then the statement ‘a and b’ is true if both a and b are true.
Rule 2. Statements with ‘OR’
If a and b are mathematical statements, then the statement ‘a or b’ is true if
- Case 1: By assuming that a is false, show that b must be true.
- Case 2: By assuming that b is false, show that a must be true.
Rule 3. Statements with ‘If-Then’
To prove the statements ‘if a then b’ consider
- Case 1: By assuming that a is true, show that b must be true.
- Case 2: By assuming that b is false, show that a must be false (Contrapositive Method).
Rule 4. Statements with ‘If and Only If’
To prove the statements ‘a if and only if b’ consider
- If a is true, then b is true, and
- If b is true, then a is true.
Rule 5. By Contradiction
To check a statement a is true, we assume that a is not true i.e., a is false or ~a is true. This assumption will contradict the validity of the statements so formed. Hence, we can say that a is true.
Laws of Algebra of Statements
-
Idempotent Laws
If a is any statement then
a ∧ a = a
a ∨ a = a
-
Associative Laws
Consider a, b, and c be the three statements, then
(a ∧ b) ∧ c = a ∧ (b ∧ c)
(a ∨ b) ∨ c = a ∨ (b ∨ c)
-
Commutative Laws
- If a and b are the two statements, then
a ∧ b = b ∧ a
a ∨ b = b ∨ a
-
Distributive Laws
Consider a, b, and c be the three statements, then
a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)
a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c)
-
De Morgan’s Laws
For the statements a and b
∼ (a ∧ b) = (∼a) ∨ (∼b)
∼ (a ∨ b) = (∼a) ∧ (∼b)
-
Identity Laws
For any statement a, then
a ∧ F = F
a ∧ T = a
a ∨ T = T
a ∨ F = a
-
Complement Laws
- For any statement a,a ∧ (∼a) = Fa ∨ (∼ a) = T∼(∼a) = a∼T = F∼F = T
Solved Example for You
Problem: Write the contrapositive and the converse of the following statements.
- If n is an odd number then n2 is odd.
- If a is square then all its four sides are equal.
Solution: Contrapositive statements
- If n2 is not an odd number, it is not an odd number.
- If all the sides are not equal, a is not square.
Converse statements
- If n2 is odd, then n is odd.
- If all the four sides of a are equal, then it is a square.
Question- What is a statement in algebra?
Answer- In algebra, a statement is basically sentences which are either true or false. These mathematical statements may consist of words and symbols. For instance ‘The square root of 2 is 1’ is a mathematical statement. Evidently, it is a false one.
Question- What are the types of statements?
Answer- There are two types of statements which are Open statements and Compound statements. An open statement is a sentence containing one or more variable. Thus, when specific values are given to the variable, they are called an open statement. On the other hand, compound statements are those which are a combination of two or more simple statements. Each statement in here is a component statement.
Question- What does the Negation of Statement mean?
Answer- Negation of Statement refers to denial of the statements. This is indicated by the sign (∼). Thus, we read this as ‘not’. For instance, if b: Number 5 is an odd number. The negation of b is not b, ∼ b: Number 5 is not an odd number.
Question- List any two properties of Statements.
Answer- Two properties of statements are that they are either true or false as they can’t be both. Further, ambiguous sentences are unacceptable in mathematics.
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