BASIC MATHEMATICS FOR NATURAL SCIENCES UNDERGRADUATE STUDENT TEXTBOOK

In mathematics, one frequently comes across sentences that involve a variable. For example, \(x^2 + 2x - 3 = 0\) is one such. The truth value of this statement depends on the value we assign for the variable \(x\text{.}\) For example, if \(x = 1\text{,}\) then this sentence is true, whereas if \(x = - 1\text{,}\) then the sentence is false.

Section objectives:

After completing this section, students will be able to:-

Relationship between the existential and universal quantifiers.

Now (d) is the denial of (a), and (c) is the denial of (b), on the basis of everyday meaning. Thus, for example, the existential quantifier may be defined in terms of the universal quantifier.

Now we proceed to discuss the negation of quantifiers. Let \(P\left( x \right)\) be an open proposition. Then \(\left( \forall x \right)P\left( x \right)\) is false only if we can find an individual “\(a\)” in the universe such that \(P\left( a \right)\) is false. If we succeed in getting such an individual, then \(\left( \exists x \right)\neg P(x)\) is true. Hence \(\left( \forall x \right)P\left( x \right)\) will be false if \(\left( \exists x \right)\neg P(x)\) is true. Therefore the negation of \(\left( \forall x \right)P\left( x \right)\) is \(\left( \exists x \right)\neg P(x)\text{.}\) Hence we conclude that

Similarly, we can easily verified that

Quantifiers Occurring in Combinations

In general, from the meaning of the universal quantifier it follows that in an expression \(\left( \forall x \right)\left( \forall y \right)P(x,y)\) the two universal quantifiers may be interchanged without altering the sense of the sentence. This also holds for the existential quantifies in an expression such as \(\left( \exists x \right)\left( \exists y \right)P(x,y)\text{.}\)

In the statement \(\left( \forall x \right)\left( \exists y \right)P(x,y)\) , the choice of \(y\) is allowed to depend on \(x\) - the \(y\) that works for one \(x\) need not work for another \(x\text{.}\) On the other hand, in the statement \(\left( \exists y \right)\left( \forall x \right)P(x,y)\text{,}\) the \(y\) must work for all \(x\text{,}\) i.e., \(y\) is independent of \(x\text{.}\) For example, the expression \(\left( \forall x \right)\left( \exists y \right)(x < y)\text{,}\) where \(x\) and \(y\) are variables referring to the domain of real numbers, constitutes a true proposition, namely, “For every number \(x\text{,}\) there is a number \(y\text{,}\) such that \(x\) is less that \(y\text{,}\)” i.e., “given any number, there is a greater number.” However, if the order of the symbol \(\left( \forall x \right)\) and \(\left( \exists y \right)\) is changed, in this case, we obtain: \(\left( \exists y \right)\left( \forall x \right)(x < y)\text{,}\) which is a false proposition, namely, “There is a number which is greater than every number.” By transposing \(\left( \forall x \right)\) and \(\left( \exists y \right)\text{,}\) therefore, we get a different statement.

The logical situation here is:

Finally, we conclude this section with the remark that there are no mechanical rules for translating sentences from English into the logical notation which has been introduced. In every case one must first decide on the meaning of the English sentence and then attempt to convey that same meaning in terms of predicates, quantifiers, and, possibly, individual constants.