9.5.1. Existential Quantification Example

I want to represent a statement that every student in course number CS201 has studied calculus. If you are wondering what is this CS201 well at my institute IIIT, Bangalore the course number for discrete maths course is CS201 and say my domain is the set of all students in a college. So since I am considering for instance IIIT, Bangalore, my domain is the set of all students in IIIT, Bangalore but it could be any domain.

In this chunk, the professor is setting the context by defining the statement to be represented: 'Every student in course CS201 has studied calculus.' The domain here refers to the group of individuals we're discussing, which is the set of all students at the professor's college. By mentioning the course number 'CS201', the professor illustrates the specific context where this statement is applicable.

Imagine a school where all the students taking a math class, known as Math 101, are required to pass a prerequisite course in algebra. If we say, 'Every student in Math 101 has passed algebra,' we are making a similar assertion about a specific group of students.

So when I am saying that every student in my domain who is enrolled for CS201 has studied calculus the interpretation of that is that I am making a universal statement, a universally quantified statement where I am saying that all for every student x in my domain, if student x has enrolled for CS201, then student x has studied calculus.

This chunk clarifies that the statement regarding students and calculus is interpreted as a universal quantification, meaning it applies to all students within the specified domain. The use of predicates helps structure the assertion logically: for each student 'x', if they are enrolled in 'CS201', then they must have studied calculus.

Think of it like this: if we say, 'All birds can fly,' we are making a universal claim. If we had a specific case where every bird, say sparrows, can indeed fly, then that fits our claim. Similarly, every student in our domain studying for 'CS201' must meet the criteria of having studied calculus.

So, let me first introduce a predicate here S(x) while you can use any predicate variable but I am using S(x) for my convenience. S(x) will be true if student x has enrolled for CS201 whereas C(x) will be true if student x in your domain has studied calculus else, it will be false.

In this section, the professor introduces two specific predicates: S(x) denotes enrollment in course CS201, and C(x) indicates whether a student has studied calculus. Predicates are foundational in predicate logic as they allow us to represent properties and relationships of elements (in this case, students) within a logical framework. Thus, these predicates help in formulating the logical statements.

Think of predicates like labels. For instance, if 'x' represents a specific car, we might have P(x) labeled as 'is red' and Q(x) as 'is a convertible.' If both labels apply to the same car, that gives us useful information about it, much like how we are determining student's statuses in our logic example.

I am writing down here two expressions. One expression is for all x, S(x) → C(x) this represents that for every x in the domain here domain is the set of all students in my college, if student x has enrolled for CS201, then he has studied calculus.

In this chunk, the professor outlines the logical representation of the assertion using logical notation. The expression 'for all x, S(x) → C(x)' translates into: 'For every student 'x', if they are enrolled in CS201, then they have studied calculus.' This expression captures the intended meaning of the original statement.

Imagine a company saying, 'If a person is an employee (E), then they receive a salary (S).' Here, being an employee (E) directly leads to receiving a salary (S), similar to how enrollment in CS201 leads to having studied calculus.

Now an interesting question here is whether the statement that I want to represent is represented by the first expression or is it represented by the second expression? Very often students do think that it is the second expression which is representing the statement every student in CS201 has studied calculus but that is not the case.

The discussion in this chunk highlights a common misconception. The professor points out that many may mistakenly believe that the second expression, which states 'for all x, S(x) ∧ C(x)', correctly represents the original statement. However, this expression incorrectly implies that all students have enrolled in CS201 and have studied calculus, which is not what we are asserting. The careful delineation of conditions is critical in logic.

Imagine a scenario where you say, 'All people who have a driver's license drive a car.' The correct interpretation is, if a person has a license, they drive, but it doesn’t mean all people drive. If you say, 'All drivers have licenses', that would be a misinterpretation.

Now, let us consider the two expressions our goal is to identify whether it is the expression number one or expression number two which represents the assertion that every student in CS201 has studied calculus.

In this example, the professor runs through a specific case with students Ram, Shyam, and Balram to validate which logical expression accurately represents the assertion. By checking the truth values of the statements, he illustrates that expression one holds true in the example’s context whereas expression two does not, thereby reinforcing the importance of proper logical representation.

It's like testing a recipe: if you claim 'every chef uses fresh ingredients', you need to show every chef truly uses them. If you find even one chef doesn't, it's clear that claim doesn't hold. Thus, using real examples to validate logical statements helps solidify our understanding.

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