Thread: Condition/Truth Table
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Old October 30th, 2009, 11:55 AM
Soroban Soroban is offline
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Hello, melwin!

Quote:
Are the 2 conditions equivalent? Explain why.

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

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

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

Its negation is: .\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: .\sim(a\text{ is less than }b)\:\text{ and }\sim(a - b)

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

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


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



Quote:
\text{(b)}\;\text{Not }(a \leq b)

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

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



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

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

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


Quote:
\text{(b) }\;(a>b) \wedge (c \neq d)

\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)


Quote:
\text{(c) }\;(a=b) \vee(a=c)

\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)
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