Mathematical Statements

Introduction

The language used in maths needs to be very precise. This means that whenever there is a maths problem, every question has a right and wrong answer. In other words, the answer to any maths question that is solved or numerically calculated is a mathematical statement. This is because the result can be either correct or incorrect, true or false. 

Mathematical statements form the basis of logical reasoning in maths. Mathematical statements are also called propositions.

For example, Amit has 5 apples, and Chanda has 6 balls. What is the total number of items shared between them? The correct answer to this mathematical problem of 6+5=11. This is a mathematical statement.

Proposition in Maths

Maths propositions form the basic foundation of mathematical logic and reasoning. A mathematical proposition is made up of a mathematical statement that has either a true or a false result based on specific logic. Naturally, propositions in maths cannot have both true and false values at once. In other words, a mathematical proposition cannot be logically correct and incorrect at the same time. 

For instance, consider the following statements:

1. The sum of two positive integers ‘a’ and ‘b’ will always yield a positive integer. 
2. Triangles have three equal sides.
3. The food Alice cooked for dinner yesterday tasted delicious.

The first statement is a mathematical proposition, which is a valid mathematical statement since it has one correct answer. It is true that upon addition, two positive integers will always yield a positive integer. The second statement above is not a mathematical proposition since it has no logical basis. Triangles may or may not have equal sides, although they are required to have three sides. Only the equilateral triangle has three equal sides and angles. An alternative to this statement that is logical would be all equilateral triangles have equal sides. The third statement mentioned above is not a valid proposition in maths because there is no way to calculate or quantify the taste of food. Hence, it is not a mathematical statement. 

It is clear from the above set of examples that propositions in maths do not have more than one true value, and there must be a well-defined logical basis to derive the true value of the given mathematical statement. Certain propositional variables and symbols are used as mnemonics to represent proportions in maths. For example, ‘p → q’ is a manner in which logical statements are denoted in maths. Here, p and q are logical variables, while the arrow denotes a logical operation. 

Conditional Statement in Mathematics

Conditional statements in maths are complex propositions that are formed with two or more mathematical statements or propositions clubbed together to exhibit a certain logical characteristic. Conditional statements in maths are usually compound statements. This means that the statements often use logically connecting words such as and, or, if, then, else, either, or, neither, among and others. These are mathematically appropriate and have a logical basis for their use. 

For instance, consider the following set of propositions:

P: f(a) is greater than f(b)

Q: f(a) is less than f(c)

Z: f(b) less than f(c) 

In the above set of mathematical statements, P and Q represent two simple statements that are clubbed together based on their truth value to form another true statement, Z. This statement Z is a compound statement and is called a conditional statement in maths.

There are a few important rules that you must keep in mind while solving such logical reasoning-based conditional statements in maths. These are as follows:

1. A conditional statement that uses the word ‘and’ to connect the two propositions is said to be true only if both the component statements are true.   
2. A conditional statement that uses the word ‘and’ to connect the two propositions is said to be false if either of the component statements is false. 
3. A conditional statement that uses the word ‘or’ to connect the two propositions is said to be true if either of the component statements is true and if both conditional statements are true. 
4. A conditional statement that uses the word ‘or’ to connect the two propositions is said to be false only if both of the component statements are false. 

The Use of Quantifiers in Mathematical Statements

In maths, quantifiers are words that elevate the given proposition for every given condition. These are words such as ‘There exists’ and ‘For all’. When such words are used in a mathematical proposition, the logic of the statement is applicable through every condition. 

For example, consider the following set of mathematical statements:

1. For all given values of A, B is greater than A by a value of 10. 
2. There exists a polygon with four equal sides and four equal vertices.
3. There exists a story of a master magician that is fit for all. 

In the above system of statements, the use of ‘For all’ in the first case ensures that we logically arrive at the answer that all the values of B will be 10 more than the consequent value of A. In other words, B=A+10, which is a valid mathematical statement. The second statement is also a logical proposition since it aids in deducing the only type of polygon that meets the conditions, i.e., a square. The third statement is not a valid mathematical statement since any given logic cannot quantify it.  

Mathematical Statement Examples

• Classify each of the sentences below as an atomic statement, and molecular statement, or not a statement at all. If the statement is molecular, say what kind it is (conjunction, disjunction, conditional, biconditional, negation).
  • The sum of the first 100 odd positive integers.

This is not a statement; it does not make sense to say it is true or false.

• Everybody needs somebody sometime.

This is an atomic statement (there are some quantifiers, but no connectives).

• We can have donuts for dinner, but only if it rains.

This is a molecular statement, a conditional.

• Every natural number greater than 1 is either prime or composite.

This is an atomic statement. Even though there is an “or” in the statement, it would not make sense to consider the two halves of the disjunction. This is because we quantified over the disjunction. In symbols, we have ∀ x (x>1→(P(x)∪C(x))). If we drop the quantifier, we are not left with a statement, since there is a free variable.

• This sentence is false.

This is not a statement, although it certainly looks like one. Remember that statements must be true or false. If this sentence were true, that would make it false. If it were false, that would make it true. Examples like this are rare and usually arise from some sort of self-reference.

2.

Solution:

In the above set of equations, the → represents the dependency of ‘q’ on ‘p’. In terms of language, it can be understood as “All triangles are polygons with three sides and three vertices.” This may or may not always be true. 

• How to write a maths statement?

A mathematical statement consists of two parts. First is the hypothesis or assumptions, and the second is the conclusion. Furthermore, most of the mathematical statements you will see in first-year courses have the form “If A, then B” or “A implies B” or “A → B.”

• What is a statement in a proof?

A proof refers to a logical argument that demonstrates a specific statement, proposition, or mathematical formula is true. In addition, it contains a set of assumptions (known as axioms) connected by statements of deductive reasoning (called an argument) to drive the proposition that is being proved.

1. What is the negation of the following mathematical propositions P AND Q?

P: Miami is a big city

Q: Miami is in the United States of America

Solution:

To arrive at the answer, we first need to find out the positive synthesis of P AND Q, which is Miami is a big city in the United States of America. 

 Further negation of P AND Q:

Miami is not a big city in the United States of America.

Note: Be careful not to use ‘not’ twice in the above answer. 

Conclusion

Above all, mathematical statements are fundamental to grasping the overarching concept of logical and mathematical reasoning. It is one of the most important subjects for CBSE as well as for various entrance examinations. 

In this respect, the validity of mathematical propositions and the use of conditional statements, quantifiers, and logical operators are crucial for you to score great marks in your exams.