Question 1

We find the negation of each statement. The negation is true when the original statement is false.

Step 1 — Simple negations

Let's start with simple statements. We add "not" to negate them.

(i) The statement is "Man is mortal". Its negation is: Man is not mortal.

(ii) The statement is "Line $l$ is parallel to line $m$". Its negation is: Line $l$ is not parallel to line $m$.

(iii) The statement is "This chapter has many exercises". Its negation is: This chapter does not have many exercises.

(ix) The statement is "2 divides the positive integer $a$". Its negation is: 2 does not divide the positive integer $a$.

(x) The statement is "Integers $a$ and $b$ are coprime". Its negation is: Integers $a$ and $b$ are not coprime.

Step 2 — Negating "All" statements

When a statement says "All X are Y", its negation is "Some X are not Y".

(iv) The statement is "All integers are rational numbers". Its negation is: Some integers are not rational numbers.

Step 3 — Negating "Some" statements

When a statement says "Some X are Y", its negation is "No X are Y".

(v) The statement is "Some prime numbers are odd". Its negation is: No prime numbers are odd.

Step 4 — Negating "No" statements

When a statement says "No X are Y", its negation is "Some X are Y".

(vi) The statement is "No student is lazy". Its negation is: Some students are lazy.

Step 5 — Negating "Some... not" statements

When a statement says "Some X are not Y", its negation is "All X are Y".

(vii) The statement is "Some cats are not black". Its negation is: All cats are black.

Step 6 — Negating existence statements

The statement says something does not exist. Its negation says it does exist.

(viii) The statement is "There is no real number $x$, such that $\sqrt{x} = -1$". Its negation is: There is a real number $x$, such that $\sqrt{x} = -1$.

Answer

(i) Man is not mortal. (ii) Line $l$ is not parallel to line $m$. (iii) This chapter does not have many exercises. (iv) Some integers are not rational numbers. (v) No prime numbers are odd. (vi) Some students are lazy. (vii) All cats are black. (viii) There is a real number $x$, such that $\sqrt{x} = -1$. (ix) 2 does not divide the positive integer $a$. (x) Integers $a$ and $b$ are not coprime.