Atomic Propositions in Discrete Mathematics

The atomic proposition is a type of statement, which contains a truth value that can be true or false. For example:

This example has three sentences that are propositions. Where the first sentence is False or invalid, and the last two sentences are True or Valid.

Now we explain some sentences that are not propositions that mean they have more than one truth value or don't have a truth value.

For example:

This example has three sentences that are not propositions because the first sentence may be false or true because the value of 'a' is not specified, so we can't say that it is true or false unless we specify the value and the last two sentences don't have a truth value.

We can easily determine that any given sentence is a proposition or not by prefixing those sentences with

After putting this, we will check that the sentence is making any grammatical sense or not. Now we will use propositional variables to avoid the writing of proposition in full. The propositional variable can be represented by the lower case letters such as x, y, z, etc. If we define a propositional variable rather than a full sentence, we must define it to write something like:

There is also an alternative. We can write a sentence something like wing_flaps_are_up so that the meaning of propositional variables becomes obvious.

Connectives

Now we will explain the connective, which will help us to create complex propositions. The various connectives are shown as follows:

1. AND (∧)
2. OR (∨)
3. Negation/ NOT (¬)
4. Implication / If-then (⊃ or →)
5. Iff or If and only if (⇔)

AND(∧):

If we combine two propositions with the word 'and' and form a third proposition, it will be known as conjunction/and of original propositions. Suppose X and Y are two variables of atomic propositions. The proposition of these variables will be true if X and Y both are true. The conjunction of X and Y is shown as follows:

The operation with AND (∧) connectivity can be summarized in a truth table. The truth table idea for some formulas shows that the behavior of the formula can be described under all possible interpretations of primitive propositions included in the formula.

Suppose some formula contains n different atomic propositions. In that case, the truth table will contain 2ⁿ different lines for that formula. The reason is that every proposition can take only one value among the two values. They can be either true or false.

We will denote F for false and T for true. Now the truth table for X ∧ Y is described as follows:

Example:

We will assume the conjunction of proposition X = "Today is a sunny day" and Y = "it is Thursday today". The X ∧ Y is "Today is a sunny day and it is Thursday today". This proposition will be true only on sunny Thursday, and it will be false on any other sunny day or when it does not sunny on Thursday.

OR(∨)

If we combine two propositions with the word 'or' and form a third proposition, it will be known as a disjunction of original propositions. Suppose X and Y are two variables of atomic propositions. The proposition of these variables will be true if X and Y both are true, or either X is true, or Y is true. The disjunction of X and Y is shown as follows:

The truth table for X ? Y is described as follows:

Example:

We will assume the disjunction of proposition X = "Today is a sunny day" and Y = "it is Thursday today". The X ∨ Y is "Today is a sunny day or it is Thursday today". This proposition will be true on a sunny day or any day that is Thursday, and it will be false when it is not sunny and the day is not Thursday.

Implication or If-Then(→)

In mathematics, many sentences contain the form

We can say this in another way which says

There is a connective in propositional logic, which is if-then. The purpose of this connective is to combine two propositions into a new proposition known as implication or condition of original propositions, which is used to capture the sense of such statement.

Suppose X and Y are two variables of atomic propositions. The proposition of these variables will be true in all cases except when X is true and Y is false. The implication of X, and Y is shown as follows:

The truth table for X → Y is described as follows:

The if-then operator(→) is extremely important and is hardest to understand. So we can understand X → Y in a way that if X is false, then X → Y will be true no matter the Y's value.

Example:

We will assume the proposition X = "Today is a sunny day" and Y = "it is Thursday today". The X → Y is "If it is a sunny day then it is Thursday today". This proposition will be true if it is not a sunny day or if it is a sunny day and it is Thursday, and it is false when it is a sunny day, but it is not Thursday.

If and only if (⇔)

Here, X ⇔ Y is known as bi-conditional logical connectives. In mathematics, there is one more form of statement which says

We can understand the sense of above statement by using the bi-conditional operator (⇔). Suppose X and Y are two variables of atomic propositions. The bi-conditional of these variables will be true if X and Y are both the same. That means X and Y both are false, or X and Y both are true. The bi-conditional of X and Y is shown as follows:

The truth table for X ⇔ Y is described as follows:

The X and Y will be known as logically equivalent if X ⇔ Y is true.

Example:

We will assume the proposition X = "Today is a sunny day" and Y = "it is Thursday today". The X ⇔ Y is "If it is a sunny today if and only if it is Thursday today". This proposition will be false when it is not a sunny day, or it is not Thursday, and it is true if it is not a sunny day and it is not Thursday or if it is a sunny day and it is Thursday.

Not (¬)

All the above considered four connectivity is binary because all these four connectives have taken two arguments. Now we will consider the final connectivity 'not', which is unary because it takes only one argument.

We can prefix any proposition with the word 'not' and form a second proposition, which will be known as a negation of original propositions. Suppose X is an atomic proposition. The proposition of this variable will be true if X is false. The negation of X is shown as follows:

The truth table for ¬X is described as follows:

Example:

We will assume the disjunction of proposition X = "Today is Thursday". The negation of X will be "Today is not Thursday".

Tautology, Consistent and Inconsistent

If we have a formula, we can't tell that it is true or false without using the truth table. We usually require truth values of the component atomic propositions to find out that the given formula is true or not.

Valuation:

A valuation is a type of function used to provide the truth value of each primitive proposition. Using the valuation, it can find that any formula is true or false.

For example: Suppose there is a valuation v, such that:

Now we will evaluate the truth value of (x ∨ y) → z like this:

Through the truth table of or (∨) connectivity, we know that F ∨ T = T. That's why line 3 is justified. Through the truth table of implication (→), we know that T → F = F.

Tautology

A formula will be known as tautology if it is true under every valuation.

For example: We have to prove that [(X → Y) ∧ X] → Y is a tautology. The truth table for this formula is described as follows:

Using the above truth table, we have proved that [(X → Y) ∧ X] → Y is True. That is why it is Tautology.

Consistent or Contingency

A formula will be known as consistent if it is true under at least one valuation.

For example: We have to prove that (X ∨ Y) ∧ (¬X) is consistent. The truth table for this formula is described as follows:

Using the above truth table, we have proved that (X ∨ Y) ∧ (¬X) has both true and false. That is why it is Consistent.

Inconsistent or Contradictions

A formula will be known as inconsistent if it is false under every valuation.

For example: We have to prove that (X ∨ Y) ∧ [(¬X) ∧ (¬Y)] is consistent. The truth table for this formula is described as follows:

Using the above truth table, we have proved that (X ∨ Y) ∧ [(¬X) ∧ (¬Y)] is false. That is why it is Inconsistent.

Propositional Equivalence

Suppose we have two statements, X and Y. They will be logically equivalent to each other if it holds any of the following two conditions:

• In the first condition, the truth table of each statement will have the same truth values.
• In the second condition, the statements of bi-conditional X⇔Y will be tautology.

For example: We have to prove that ¬(X ∨ Y) and [(¬X) ∧ (¬Y)] are equivalent. The truth table for the above statement by using the first method is described as follows:

In the above truth table, we can see that both the statements ¬(X ∨ Y) and [(¬X) ∧ (¬Y)] are the same. So we can say that the statements are equivalent.

Now we will test both the statements by using the second method, which is bi-conditional.

In the above truth table, we can see that [¬(X ∨ Y)] ⇔ [(¬X) ∧ (¬Y)] is a tautology because it is true for every value of its propositional variables. So we can say that the statements are equivalent.