Universal and Existential Quantification

    In Week 3, the class covered notes on quantifiers. Fortunately, all of the logic was already taught during the first 2 weeks of MAT137. Hence, following along with the class was easy breezy. 

    The first topic being talked about was universal quantification, represented by the symbol ∀. This ‘upside down A ‘ symbol is defined as “for all” or “for every”. When being used in a statement, ∀ can either make all of the examples in the statement true or make all of the examples false. For example, the statement “all even numbers are real numbers” can be written as ∀x∈{even numbers}, x∈R, which is true. Contrarily, the statement “every real number is even,” (∀x∈R, x∈{even numbers}) is false, since odd numbers and rational numbers also belong to the set of real numbers.

    The second topic being discussed is existential quantification, represented by the symbol ∃. This “flipped E” means “there exists” or “there is at least one.” It is used in statements where there are some examples that are true or some examples are false. For instance, the statement ‘there exists an integer that is greater than zero’ can be written as “∃x∈Z, x>0”. This statement is true because it is basically saying “there are some positive integers”, meaning that there could also be negative integers (a proof for this would be the number 3). Now, if I change the existential quantifier in this statement to the universal quantification so that “all integers are positive” , then my statement would be false since a counter-example would be that -2 is also an integer but it is smaller than 0. Differently, if I say “there is an integer between the number 1 and 2” (∃x∈Z, 1<x<2), then this statement is false since that no integers are between 1 and 2.

    Lastly, the two quantifiers can be combined into a more complex statement. For instance, “for all integers, there exists at least one other integer that is greater than the said integer” (∀x∈Z, ∃y∈Z, x<y) or in simpler words “a biggest integer does not exist”, which is true. However, if I switched the order of the statement and turn it the other way around into “∃y∈Z, ∀x∈Z, x<y”, then the statement is false, since it is basically saying that “there exists one integer that is bigger than every other integer”. In conclusion, the order in which an existential or universal quantification is used is very important, and if applied in the wrong way would make a true statement false.