Question

Negation of a statement 'IF $\forall$ x, x is a complex number, then x² <0' is [2022]

• A. $\exists$x, x is not a complex number and x² $\geq$0.
• B. $\forall$x, x is a complex number and x² <0.
• C. $\exists$x, x is not a complex number and x² <0.
• D. $\forall$x, x is a complex number and x² $\geq$0.

Answer 1

Answer: (D)

$\forall$x, x is a complex number and x² $\geq$0.

Answer 2

Let

$p:\forall x,\; x\text{ is a complex number} \quad\text{and}\quad q: x^2 < 0$.

The given statement is of the form

$p\to q$.

The negation of an implication is given by:

$\neg(p\to q) \equiv p \land \neg q$.

So we have:

$\neg(p\to q) \equiv \Bigl[\forall x,\; x\text{ is a complex number}\Bigr] \land \Bigl[ \neg\bigl(x^2<0\bigr)\Bigr]$.

Since

$\neg(x^2 < 0) \equiv x^2 \ge 0$,

the negation becomes:

$\forall x,\; x \text{ is a complex number and } x^2\ge 0$.