 # 3. Logical conjunction Por: Sergio Cohaguila

Amor a la física y matemáticas

Here again in this new and third section of the chapter on propositional logic. In this opportunity, we will discuss one of the first logical connectives with its clear definition and some illustrative examples, I mean the logical conjunction.

It is a logical connective widely used in set theory, especially in the definition of the intersection between two or more sets, it can be said that it is a connective of conditions since it must meet certain simultaneous requirements to consider the truth value of an argument. Let’s start studying it.

## What is the logical conjunction?

The logical conjunction has the property of adding mandatory conditions through the predicate applied to the subject, for example, if we want Pablo to be a bricklayer, but in turn, we want Pablo to also be a student, we will use the logical connective “and”, writing like this ” Pablo is a bricklayer and student”.

In this case, if these two conditions are taken into account for “Paul,” we must also take into account the truth values of those conditions.

That is, if “Pablo is a bricklayer” is true and it is also true that “Pablo is a student”, therefore, the proposition “Pablo is a bricklayer and a student” is also true.

Now we will see the meaning of the logical conjunction by means of the following definition based on this explanatory example, then we will move on to the examples.

### Definition of logical conjunction

The logical conjunction with symbol $$\wedge$$ is a logical connective that connects two statements $$p$$ and $$q$$ forming a new statement $$p \wedge q$$ such that its validity turns out to be true if the statements $$p$$ and $$q$$ are true and false if at least one of these statements is true.

Statements of the type “$$p \wedge q$$” from now on we will call it conjunctive proposition. In propositional logic we will omit the meaning of the arguments of the statements, although they are used as examples to explain the logic of this connective, what prevails more is only the truth values of the statements and not their meaning since it is the only thing that matters to propositional logic.

Therefore, the logical rule of conjunction is always true if the propositions that make it up are always true, otherwise, it is always false.

That is why the concept of the logical conjunction is related to the intersection in the elementary theory of sets since it must simultaneously fulfill two conditions, if it does not fulfill even one condition, then there is no intersection.

Examples

Let the following statements be:

1. The cat is a feline and the dove is a bird.
2. iPhones are smartphones and are made by Samsung.

For statement 1, it can be broken down like this:

• The cat is a feline.
• The dove is a bird.

Exactly these two simple sentences are true and by the definition of the logical conjunction, we say that sentence 1 is true, let’s go to the next sentence.

For statement 2, we can separate it like this:

• iPhones are smartphones.
• The iPhone is manufactured by Samsung.

The first simple statement is true because an iPhone has all the characteristics of a smartphone, but the second is false because iPhones are manufactured by the multi-million dollar company Apple and not by Samsung.

Therefore, by the definition of the logical conjunction, it is enough that a component proposition is false for statement 2 to be completely false.

Sentences 1 and 2 of the compound statements in example 2 can be written as follows:

• $$p \wedge q$$

Where sentence 1 can be considered like this:

• $$p$$ = The cat is a feline. (true)
• $$q$$ = The dove is a bird. (true)

and for the case of sentence 2, it would be:

• $$p$$ = iPhones are smart phones. (true)
• $$q$$ = iPhones are from Samsung. (false)

According to the definition of the logical conjunction, the truth value of sentence 1 would be true, symbolically:

$\mathrm{V} ( p \wedge q) = V$

and for sentence 2 it would be false, symbolically:

$\mathrm{V} (p \wedge q) = F$

## Truth table

With the previous examples, the behavior of the conjunctive connectives is clear and from there we can design a truth table according to the definition given for two statement $$p$$ and $$q$$ where we will find 4 combinations of their truth values of $$p \wedge q$$. The table is:

$\begin{array}{ c | c | c } p & q & p \wedge q \\ \hline V & F & F \\ V & F & F \\ F & V & F \\ F & F & F \end{array}$

## Some laws of logical conjunction

The logical conjunction also contemplates a series of properties but together with the logical negation and the rest of the logical connectives that we will see later, however, we will see some of them right now:

Here I will only present some of these laws, be the statements $$p$$, $$q$$ and $$r$$, we have:

• Associative law: $$p \wedge q \wedge r = p \wedge ( q \wedge r )$$.
• Existence of the neutral element: $$\mathrm{V} (p) \wedge V = V$$.
• Commutative law: $$p \wedge q = q \wedge p$$.
• Existence of the complementary element: $$\mathrm{V} ( \sim p \wedge p ) = F$$. It means that a statement cannot contradict itself.

To mention just a few.

Another important point, the negation of a conjunction is not another conjunction, that is, the negation of $$p \wedge q$$ is not $$\sim p \wedge \sim q$$, its negation turns out to be equivalent to an in a statement formed by a logical disjunction, that is $$\sim (p \wedge q ) = \sim p \vee \sim q$$, but we will see this in the next section, let’s see some examples.

Examples

• We are all free and happy.
• Julieta is beautiful and should be a model.
• They are all criminals but with bipolar disorder.
• Jimena is on the beach and is eating.
• Dogs have a tail, paws, and muzzle.

As you can see in these examples, both the letter “and” and the word “but” and even commas depending on the context can be represented by the logical conjunction, for the dog example we can write it as $$p \wedge q \wedge r$$.

## The conjunction in set theory

Logical connectives can build the elementary aspects of set theory, such as union, intersection, etc. in this case, the logical conjunction in the domain of the sets represents the intersection.

For example, if an element $$x$$ is in the set $$\mathrm {A}$$ symbolized by $$x \in \mathrm {A}$$, but it is also in the set $$\mathrm {B}$$ symbolized by $$x \in \mathrm {B}$$, we use the logical conjunction symbol “$$\wedge$$” to indicate that it is simultaneously in a $$\mathrm {A }$$ and $$\mathrm {B}$$, like this:

$x \in \mathrm{A} \wedge x \in \mathrm{B}$

But at the level of set theory, its symbol is “$$\cap$$”, and it is represented like this:

$x \in \mathrm{A \cap B }$

But to unify both the conjunction and the intersection in the same sentence, we will write it like this:

$\mathrm{ A \cap B } = \left \{ x | x \in \mathrm{A} \wedge x \in \mathrm{B} \right \}$

Graphically it is represented like this:

Later on, we will adequately develop all elementary set theory since we will use the concepts of propositional logic to demonstrate some properties of elementary set theory.

In this way we finalize the basic theory of logical conjunction, the next section we will discuss logical disjunction and its variant, exclusive disjunction. And that’s all friends, see you in the next session, greetings.