Test: Mathematical Statements - JEE MCQ

Cube of an integer is +ve or -ve.
p: Cube of an integer is +ve
q: Cube of an integer is -ve

p: A rectangle is a quadrilateral.
q: The opposite sides of a rectangle are equal.
Compound statement using connecting word: ‘and’
A rectangle is a quadrilateral and its opposite sides are equal.

 Rules regarding the use of connective “Or”
If all of the component statements connected by the connective and is false then the entire statement is false.

In mathematical reasoning, a statement is called a mathematically acceptable statement if it is either true or false but not both. In addition, each of these statements is termed to be a compound statement. Furthermore, the compound statements are combined by the word “and” (^) the resulting statement is called conjunction denoted as a ^ b.

Quantifier is “For Every”
Negotiation of statement is : There exist a real number x such that x is not less than x+5

Since there are 28 or 30 or 31 days in a month
Its a false statement
Hence it is a statement.

1. Negate the quantifier
   Original: “For every real number x, x³ > x².”
   Negation: “There exists a real number x such that (not) x³ > x².”

2. Negate the inequality
   “Not (x³ > x²)” is “x³ ≤ x².”
   (We must include equality because “>” fails both when it is “=” and when it is “<”.)

So the negation is: There exists a real number x such that x³ ≤ x².

3. Quick algebra check (why this is true)
   x³ ≤ x² ⇔ x²(x − 1) ≤ 0.
   Since x² ≥ 0 for all real x, this inequality holds whenever x − 1 ≤ 0, i.e., for all x ≤ 1 (and it’s also true at x = 0 because the product is 0).
   Examples:
   • x = 0 → 0³ = 0² (equality), so “>” fails.
   • x = 1 → 1³ = 1² (equality), so “>” fails.
   • x = 1/2 → (1/8) < (1/4), so “>” fails.

Therefore, the original statement “for every x, x³ > x²” is false, and its correct negation is exactly: There exists a real number x such that x³ ≤ x².

Component statements of ‘All integers are either even or odd’ are:
P: All integers are even
Q: All integers are odd

Mathematics is difficult for some but not for other.
Hence this statement can be true or false.
So it is not a statement.