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.
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
Conditional statements express the relationship between two propositions using the if-then format.
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
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
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
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
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
The following examples clearly illustrate how we write the above-mentioned types of statements.
Example | Type of statement |
For every natural number n, it’s true that n is greater than zero. | Universal |
If the condition is that it rains, then the consequence is that the ground will be wet. | Conditional |
There exists at least one letter a in the alphabet, and it’s a vowel. | Existential |
For all real numbers x, if x is greater than 5, then 2x is greater than 10. | Universal Conditional |
For all countries c, there exists a city c1 that is the capital of c. | Universal Existential |
There exists a prime number p such that for all natural numbers n, np is divisible by n. | Existential Universal |
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.
Example Statements | Simple, Compound or Neither |
The sky is blue. | Simple |
Cats and dogs are mammals. | Compound |
5 + 5 = 10 | Simple |
What time is it? | Neither |
Either the sun is shining, or it’s raining, and if it’s raining, then I’ll stay indoors. | Compound |
She likes chocolate and vanilla ice cream. | Compound |
Don’t be late! | Neither |
He’s a doctor. | Simple |
Now, test your understanding of these various statements by attempting the quiz below.
What does the following statement mean?
∀x ∈ real numbers, if x > 5, then 2x > 10
For all real numbers, 2x is always greater than 10.
If 2x is greater than 10, then x must be greater than 5.
For all real numbers greater than 5, 2x is greater than 10.
If x is greater than 5, then 2x is greater than 10.