What are the types of statements in discrete structures?

In the study of discrete structures, mathematical statements are used to express relationships, make assertions, and construct logical arguments. These statements come in different forms and serve various purposes, ranging from making universal claims about entire sets to specifying conditional relationships.

Let’s explore these statements in detail.

Types

1. Universal statements quantify all the elements in a specific set and make assertions about them. The format for universal statements is:

This means that for every element $x$ in the set $S$, the property $P(x)$ holds true.

2. Conditional statements express the relationship between two propositions using the if-then format.

   1. If-then statement is a type of conditional statement. It expresses a cause-and-effect relationship where the if proposition leads to the then proposition. Its format is:

This means that if condition $P$ is true, then the consequence $Q$ must also be true.

3. Existential statements assert the existence of at least one element in a specific set that satisfies a given property. Its format is:

This means that there exists an element $x$ in the set $S$ for which the property $P(x)$ holds true.

4. Universal conditional statements combine universal quantification and conditional logic, asserting that a relationship exists between two propositions for all the elements in a specific set. Its format is:

This means that for all elements $x$ in the set $S$, if the condition $P(x)$ is satisfied, then the consequence $Q(x)$ must be true.

5. Universal existential statement combines universal quantification and existential quantification, asserting that a certain condition holds true for all the elements in a specific set. Its format is:

This means that for each elements $x$ in the set $S$, there exists an element $y$ in the set $T$ such that the property $P(x, y)$ holds true.

6. Existential universal statement combines existential quantification and universal quantification, asserting the existence of an object that satisfies a given condition and stating that all objects of the same kind also meet this condition. Its format is:

This means that there exists an element $x$ in the set $S$ such that for all elements $y$ in the set $T$, the property $P(x, y)$ holds true.

The following examples clearly illustrate how we write the above-mentioned types of statements.

A simple statement is a declarative sentence that’s either true or false but can’t be both. If the sentence is true, it has a truth value of true; if it’s false, then the truth value is false. Simple statements are combined to form compound statements. Simple statements are connected using logical connectives such as ›AND, OR, and NOT.

Now, test your understanding of these various statements by attempting the quiz below.