Thread: Boolean Equivalents - Basic Identities

I'm looking at a table meant to help me simplify condition code, but I'm not exactly sure what these statements mean. I understand the first one: !!x (double-arrow) x. No problem. A double-negative equals a positive (or "true"). But the rest of the equivalencies are not making sense to me yet. Like this one:
x || false (double-arrow) x
Does that read like this: "a variable named "x" OR a false result of an expression is the same thing as that variable named "x"? If that is in fact how that reads, then I don't see how that is equivalent to "x". I'm having the same problem understanding:
x && true (double-arrow) x
and;
x || true (double-arrow) true
and so on...

Maybe this first-time poster just needs a nap. But please explain, anyway.

Norm,

This is right out of an intro to Java textbook. This reference table is titled Basic Identities of Boolean Logic. These statements are meant to show a simplified equivalent (right side of the double arrow) to the condition statement on the left side of the double arrow. The double arrow is the symbol for "equivalent". I don't have a double arrow on my keyboard, so I wrote it out. I'm not following the logic of these "equivalent" statements, though.

Does that clarify at all?

The book shows a solid line between the arrow points. It means "the same thing as" (equivalent). I take it that's what your symbol means, too.

...and yes, it look very much like your example - only with a solid line between the arrow points.

Thank you, Zaphod_b!

Great explanation!

Richard Feynman tells a story somewhere about having a cousin who was baffled by some problem in maths - a calculation of some sort. He thought that the calculation was quite straight forward and told his cousin what the answer was, but his cousin objected: "No, I have to solve it with algebra!". Feynman expresses some dissatisfaction with algebra if it acts to obscure the calculation.

Consider the following rules of algebra (the sort of thing Feynman's cousin might have found confusing). And think about what they mean:

x = -(-x)
x = x + 0
x.1 = x

It is reasonable to say that these rules express the fact that the left and right hand side have the same value, whatever the value of x might be. They are valid identities of algebra but, it must be admitted, they are pretty useless aids to calculation.

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Your book is saying something similar about thinking and the rules of logic. Variables and other expressions, like the letters and expressions of algebra, have values. And it is possible to find pairs of things that always have the same value whatever the value of x may be.

x <-> !!x
x <-> x || false
x && true <-> x
x || true <-> true

If you don't think these identities always hold, come up with a value of x that makes the two sides of the "equation" have different values.

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The algebra examples were chosen to mimic the logic ones. (With true and false playing the role of one and zero) But we notice a couple of differences. First the logical x can only have two values which makes the rules of logic rather easy to check. And secondly the last of the logic rules has no straight forward counterpart in arithmetic.

I'm inclined to agree with Feynman: these rules of logic add little to our ability of to perform boolean calculations. (They are, however, convenient symbolic shortcuts to help us reason about such calculations). So try and understand why they are true, but don't get too hung up over them: they are not expressing profound truths about reasoning.

[Edit] slow

pbrockway2, Thanks for your helpful insights.

I think the point of these basic identities, as they're called in the text, is to give guidance for simplifying code in the condition. Whether or not it will prove helpful for that, well...

They're also meant to give us alternative solutions when we're sort of stuck.

I think the point of these basic identities, as they're called in the text, is to give guidance for simplifying code in the condition.

Fair enough - and anything that helps is ... helpful!

I find it helpful to read an expression. Actually read it, rather than just letting it hit the eyes as a sequence of symbols. Often I'll mentally respond to myself "Oh!, that just means...". The best rule of thumb, though, is never to write long, ugly, unintelligible conditions. It costs nothing to break them up into component subexpressions and assign them meaningfully named variables.