Question

In the context of admission to IGNOU, give examples of the following:
i) an implication;
ii) the converse of (i) above;
iii) a two-way implication which is true;
iv) a statement involving both ∀ and ∃.
v) the contrapositive of (i) above.

Answer 1

Answer on Question #44545 – Math - Discrete Mathematics

Problem.

In the context of admission to IGNOU, give examples of the following:

i) an implication;

ii) the converse of (i) above;

iii) a two-way implication which is true;

iv) a statement involving both $\forall$ and $\exists$.

v) the contrapositive of (i) above.

Solution.

Suppose that $a, b$ in i, ii, iii, v are real numbers.

i) If $a = b$, then $a^2 = b^2$.

$$a = b \Rightarrow a^2 = b^2.$$

ii) If $a^2 = b^2$, then $a = b$.

$$a^2 = b^2 \Rightarrow a = b.$$

iii) $a = b + 1$ if and only if $b = a - 1$.

$$a = b + 1 \Leftrightarrow b = a - 1.$$

iv) For any real number $a$ exist real $b$, such that $a = b + 1$.

$$\forall a \in \mathbb{R} \ \exists b \in \mathbb{R} \colon a = b + 1.$$

v) If $a^2 \neq b^2$, then $a \neq b$.

$$a^2 \neq b^2 \Rightarrow a \neq b.$$

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