Which disjunction is always true for any real number x?
Which disjunction is always true for any real number x?
The Correct Answer and Explanation is:
Correct Answer:
The disjunction that is always true for any real number xxx is:
x<0x < 0x<0 or x≥0x \geq 0x≥0
Explanation
In logic and mathematics, a disjunction is a compound statement formed using the word “or”. It is symbolized by ∨\vee∨. A disjunction is true if at least one of the statements it connects is true.
Let’s analyze the statement:
x<0x < 0x<0 or x≥0x \geq 0x≥0
This disjunction divides the real number line into two exhaustive and mutually exclusive parts:
- x<0x < 0x<0: This includes all negative real numbers.
- x≥0x \geq 0x≥0: This includes zero and all positive real numbers.
Every real number either falls into the first category (less than zero) or the second (greater than or equal to zero). There are no real numbers that fall outside these two possibilities. That makes the disjunction always true—it covers every possible real number.
This type of statement is known as a tautology in logic: a statement that is true regardless of the truth value of its individual components. In this case, regardless of what real number xxx you choose, it will either satisfy x<0x < 0x<0 or x≥0x \geq 0x≥0, thus making the whole disjunction true.
Here’s how it works with examples:
- If x=−5x = -5x=−5, then x<0x < 0x<0 is true → the whole disjunction is true.
- If x=0x = 0x=0, then x≥0x \geq 0x≥0 is true → the whole disjunction is true.
- If x=7.2x = 7.2x=7.2, then x≥0x \geq 0x≥0 is true → the whole disjunction is true.
This logical structure ensures the disjunction is always satisfied, no matter what real number xxx you plug in. Therefore, it is universally true for the set of real numbers.
