QUANTIFIERS

Introduction

Assume you’re conversing with your buddy Mary, and she’s telling you about two clubs she’s joined. ‘There exists a member of Club 1, such that the individual has red hair,’ she explains when describing the people in the first club. ‘For all members in Club 2, the member has red hair,’ she explains when characterising the second club.

What can you tell me about the hair colour of Club 1 and Club 2 members based on these two statements? Let’s take a closer look at her statements and dissect them.

Meaning of Quantifiers

The expressions ‘there exists’ and ‘for all’ are extremely important in logic and logic assertions in mathematics. They are so fundamental that they have their name: quantifiers. Quantifiers are words, expressions, or phrases that specify how many things a statement refers to.

 “There is” and “For all” are examples of quantifiers. “There exists” is another phrase that appears in mathematical statements.

Consider the following statement:

p: There is a rectangle in which all sides are equal.

This indicates that there is at least one rectangle with equal sides.

“For every” is a term that is closely related to “there exists” (or for all). Consider the following statement. p: Pi is an irrational number for every prime number p. This indicates that if S symbolises all prime numbers, then pi is an irrational number for all members p of S.

A mathematical statement that begins with “for every” can be taken to mean that all members of the given set S where the property applies must satisfy that property. It’s worth noting that knowing exactly where a specific connecting word appears in a phrase is critical.

Compare and contrast the following two sentences:

1. There is a positive number y such that y < x for every positive number x.

2. There is a positive number y such that we have y < x for every positive number x.

These statements may appear to be similar, but they do not say the same thing. In fact, (1) is correct and (2) is incorrect. To understand a piece of mathematical writing, all the symbols must be carefully introduced, and each symbol must be introduced precisely at the proper time — neither too early nor too late.

Types of Quantifiers

There are two types of quantifiers:

• Existential Quantifier

The statement ‘there exists’ is known as an existential quantifier, and it means that at least one element exists that fulfils a given attribute. Mary told you at Club 1 that a member has red hair. This indicates that at least one club member has red hair, but not necessarily all.

• Universal Quantifier

The phrase “for all” or  “given any” is known as a universal quantifier, and it denotes that all items of a set meet a property. ‘For all members in Club 2, the member has red hair,’ Mary stated about Club 2. This indicates that all of Club 2’s members have red hair.

The following are a few instances of assertions involving quantifiers in mathematical logic:

There is an integer x for which 5 – x = 2.

2n is an even number for all natural numbers n.

The existential quantifier is used in the first sentence, indicating that at least one integer x fulfils the equation 5 – x = 2. The second statement uses the universal quantifier, which states that 2n is an even integer for every natural number n.

Notation of Quantifiers

When writing mathematical proofs, assertions and theorems, many explanations are required. As a result, mathematical notation is frequently utilised to condense lengthy explanations and provide a breather for your writing hand.

The fact that mathematical notation is the same in every language means that mathematicians can still communicate even if they don’t speak the same language. 

Both of the quantifiers have symbols that we employ. The universal quantifier is represented by an upside-down A, while a reverse E represents the existential quantifier.

When creating assertions that use these quantifiers, we can use this notation. 

Practice Quantifier examples

• Write the negation of the following statements:

(i) p: For every positive real number x, the number x − 1 is also positive

(ii) q: All cats scratch

(iii) r: For every real number, either x > 1 or x < 1

(iv) s: There exists a number x such that 0 < x < 1

Solution

I) Here is the negation of assertion p:

There exists a positive real number x for which x-1 is negative.

(ii) Here is the negation of assertion q: 

There exists a cat that does not scratch.

(iii) The negation of assertion r is: 

There exists a real number x for which neither x > 1 nor x<1 exists.

(iv) Here is the negation of statement s:

There does not exist a number x for which 0 < x < 1 is true.

• In the following sentences, find the quantifier and write the negation of the statements.

I) A number exists that is equal to its square.

(ii) x is less than x + 1 for every real number x.

(iii) Every state in India has its capital.

Solution

I) In this case, the quantifier is ‘there is.

The converse of the statement is: A number exists that is not equal to its square.

(ii) The quantifier is ‘for every’ in this case.

The negation of the assertion is: x is not smaller than x + 1 for at least one real number x.

(iii) In this case, the quantifier is ‘there is.

The converse of the sentence is: In India, there is a state without a capital.

• In the following sentences, find the quantifier and write the negation of the statements.

I) There exists a number equal to its cube. 

(ii) For every real number x, x is less than x – 1.

(iii) For every country in the world, there is a capital.

Solution

I) “There exists” is the quantifier.

This statement’s negation is as follows:

There is no such thing as a number that is equal to its cube.

(ii) “For every” is the quantifier.

This statement’s negation is as follows:

There is a real number x that is greater than or equal to x-1.

(iii) “There exists” is the quantifier.

This statement’s negation is as follows:

In the world, there is a country that does not have a capital.

Conclusion

Quantifiers are an important topic and make up a majority portion of Mathematical Reasoning. These statements generally consist of the words “There exists” and ” For every.” Both these connectivity follows some special rules which need to be learned and applied while solving questions.