1. Condition/Truth Table

Are the 2 conditions equivalent? Explain why.
a. Not(a<=b)
(a>=b) OR Not (a=b)

b. Not(a<=b)
(a>=b) AND Not (a=b)

Write a condition equivalent to the negation of the given condition, and that does not use the NOT operator.
a. a<b

b. (a>b) And (c<>d)

c. (a=b) Or (a=c)

2. Hello, melwin!

Are the 2 conditions equivalent? Explain why.

$\displaystyle \text{(a)}\;\text{ Not }(a \leq b)$

. . $\displaystyle (a \geq b)\text{ or Not } (a=b)$

$\displaystyle a \leq b$ means: $\displaystyle (a$ is less than $\displaystyle b)$ or ($\displaystyle a$ equal to $\displaystyle b.)$

Its negation is: .$\displaystyle \sim(a \leq b) \quad=\quad\sim\bigg[(a\text{ is less than }b) \text{ or } (a = b)\bigg]$

By DeMorgan's Law, this is: .$\displaystyle \sim(a\text{ is less than }b)\:\text{ and }\sim(a - b)$

. . . . . . . . . . . . . . . . $\displaystyle =\;(a\text{ is greater than or equal to }b)\;\text{ and }\:(a \neq b)$

. . . . . . . . . . . . . . . . . . . . $\displaystyle =\;(a \geq b) \text{ {\color{red}and} Not }(a = b)$

This is not what the second statement says; they are not equivalent.

$\displaystyle \text{(b)}\;\text{Not }(a \leq b)$

. . $\displaystyle (a \geq b)\text{ and Not }(a = b)$

These two are equivalent . . . see part (a).

Write a statement equivalent to the negation of the given condition
that does not use the NOT operator.

$\displaystyle \text{(a)}\;a < b$

$\displaystyle \sim(a < b) \quad\Longrightarrow\quad a \geq b$

$\displaystyle \text{(b) }\;(a>b) \wedge (c \neq d)$

$\displaystyle \sim\bigg[(a > b) \wedge (c \neq d)\bigg] \quad\Rightarrow\quad \sim(a > b)\: \vee \sim(c \neq d) \quad\Rightarrow\quad (a \leq b) \vee (c = d)$

$\displaystyle \text{(c) }\;(a=b) \vee(a=c)$

$\displaystyle \sim\bigg[(a=b) \vee (a = c)\bigg] \quad\Rightarrow\quad \sim(a=b)\: \wedge \sim(a = c) \quad\Rightarrow\quad (a \neq b) \:\wedge (a \neq c)$

3. hm

So both question 1 and 2 are true?