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Logics in Math











































The temperature is not 90 degrees if and only if the air conditioner is not working or the apartment is not hot.

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Jan 22, 2012
Logic
by: Staff

Question:

The temperature is not 90 degrees if and only if the air conditioner is not working or the apartment is not hot.


Answer:

X = temperature of 90 degrees

C = air conditioner working

H = apartment hot

“OR” = (written as ⋁) a logical operation in which either or both input values produce a “true” result

“NOT” = (written as ~) a negation

“Conditional If-Then” = (written as →) If the statement preceding this symbol is true, then the statement following this statement is also true.

“if and only if” = (written as ↔) biconditional



"T" = True

"F" = False

The purpose of logic is to enable you to construct valid arguments which satisfy your basic premise: "If all of your premises are true, then the your conclusion must also be true."

Your original problem statement is: The temperature is not 90 degrees if and only if the air conditioner is not working or the apartment is not hot.

It will be helpful to rewrite the problem statement as follows:

If and only if the Air Conditioner is not working or the apartment is not hot, then the temperature is not 90 degrees.

The Boolean expression for this statement can be written:

(~C ⋁ ~H) ↔~X

This relationship can be also expressed in a “Truth Table”.

A truth table is a list of every possible combination of input values and the output value for each combination. It is a picture of every possibility.

A “Truth Table” for your problem statement looks like this:

Truth Table

C    H     X    (~C ⋁ ~H) ↔~X

T     T     T             T


T     T     F             F

T     F     T             F

T     F     F             T

F     T     T             F

F     T     F             T

F     F     T             F

F     F     F             T





Thanks for writing.
Staff

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