## QuantifiersQuantifier is used to quantify the variable of predicates. It contains a formula, which is a type of statement whose truth value may depend on values of some variables. When we assign a fixed value to a predicate, then it becomes a proposition. In another way, we can say that if we quantify the predicate, then the predicate will become a proposition. So quantify is a type of word which refers to quantifies like There are mainly two types of quantifiers that are universal quantifiers and existential quantifiers. Besides this, we also have other types of quantifiers such as nested quantifiers and Quantifiers in Standard English Usages. Quantifier is mainly used to show that for how many elements, a described predicate is true. It also shows that for all possible values or for some value(s) in the universe of discourse, the predicate is true or not.
"x ≤ 5 ∧ x > 3" This statement is false for x= 6 and true for x = 4. Now we will compare the above statement with the following statement "For every x, x ≤ 5 ∧ x > 3" This statement is definitely false. Now we will again define a statement "There exists an x such that "x ≤ 5 ∧ x > 3" This statement is definitely true. The phrase "there exists an x such that" is known as the existential quantifier, and "for every x" phrase is known as the universal quantifier. The variables in a formula cannot be simply true or false unless we bound these variables by using the quantifier.
Suppose we have two statements that are ∀x : x. For x = 1, the first statement ∀x : x^{2} > 2^{2} +1 > 0 is true, but the second statement ∀x : x^{2} > 2 is false, because it does not satisfy the predicate. On the other side, if we write the second statement as ∃x : x 2 > 2, it will be true, because x = 2 is an example that satisfies it.In the quantified expression, if there is a variable, then we always assume that the variable comes from some base set. If we specify x as a real number, then the statement ∀x : x ## Universal QuantifiersSometimes the mathematical statements assert that if the given property is true for all values of a variable in a given domain, it will be known as the The sentence ∀xP(x) ⇔ P(n _{1}) ∧ P(n_{2}) ∧ · · · ∧ P(n_{k})
The statements can be: "Every electrical student must take an electronics course". The following syntax is used to define this statement: ∀x(Q(x) ⇒ P(x)) This statement can be expressed in another way: "Everybody must take an electronics course or be an electrical student". The following syntax is used to define this statement: ∀x(Q(x) ∨ P(x))
The statement must be: ∀x (x is a square ⇒ x is a rectangle), i.e., "all squares are rectangles.'' The following syntax is used to describe this statement: ∀xP(x) ⇒Q(x) Sometimes, we can use this construction to express a mathematical sentence of the form "if this, then that," with an "understood" quantifier. ## Universal conditional statement:This statement has the form: ∀x, if P(x) then Q(x). For example: In this example, we will rewrite the below statement in the form: ∀______, if______then______ If Jack is 18 years old or older, then he is eligible to vote. ## Existential QuantifiersSometimes the mathematical statements assert that we have an element that contains some properties. Using existential quantifiers, we can easily express these statements. The existential quantifier symbol is denoted by the If finite values such as {n ∃xP(x) ⇔ P(n _{1}) ∨ P(n_{2}) ∨ P(n_{3}) · · · ∨ P(n_{k})
This statement is false for all real number which is less than 4 and true for all real numbers which are greater than 4. This statement is false for x= 6 and true for x = 4. Now we will compare the above statement with the following statement. So ∃xP(x) is true ## Negating Quantified StatementsEarlier we have explain a example in which the statement ∀x : x is true for x = 1. The first statement is false because x =1 is unable to satisfy the predicate. In this case, we find a solution that says we can negate a ∀ statement by flipping ∀ into ∃. After that, we will negate the predicate inside.^{2} +1 > 0
If the statement predicate the negation of ∃x : P(x) is ∀x : P(x) ## Nested QuantifiersThe nested quantifier is used by a lot of serious mathematical statements. For example: Let us assume a statement that says, "For every real number, we have a real number which is greater than it". We are going to write this statement like this: ∀x ∃y : y > x Or assume a statement that says, "We have a Boolean formula such that every truth assignment to its variables satisfies it". We are going to write this statement like this: ∃ formulaF ∀ assignmentsA : A satisfies F. It is very important to understand the difference between statements that indicate ## Negating Nested QuantifiersIn the nested quantifier, we can negate a sequence with the help of flipping each quantifier in the sequence, and after that, we will negate the predicate like this: Negation of ∀x ∃y : P(x, y) is ∃x ∀y : P(x, y) When we think, we can realize that it makes sense intuitively. Now there are some sequences that are unbounded such as 1, 4, 9, 16, 25, …., and some sequences that are not, such as 1/2 , 3/4 , 7/8 , ...... If a sequence is not bounded, it means that it contains an upper-bounded x such that sequence's every number is at most x. If we want to derive this mathematically, we can do this by negating the definition of unboundedness. If ## Quantifier in Standard English UsagesWhen we notice, we will realize that quantifiers and Standard English usages are familiar to each other. "∀x in UK, x has a job" If we want to disagree with this statement, we must negate the above statement by flipping ∃ into ∀. After that, the predicate will be negated like this: We can reverse the same things by flipping ∃ into ∀. If someone says, "India has a cricket player who makes |