Examples and Explanations of Basic Properties of EqualityDate: 07/31/2004 at 15:02:12 From: Alyssa Subject: identity and equality properties Can you put these properties in simpler terms? I have no clue what they are trying to say. Substitution property of equality: for any numbers a and b, if a = b, then a may be replaced by b in any expression. Symmetric property of equality: for any numbers a and b, if a = b, then b = a. Transitive property of equality: for any numbers a, b and c, if a = b and b = c then a = c. I tried substituting numbers for the variables, but it didn't make it any clearer. Date: 07/31/2004 at 22:39:51 From: Doctor Peterson Subject: Re: identity and equality properties Hi, Alyssa. These are all so basic that it can be hard to see why we bother saying them! What mathematicians try to do is to boil down our reasoning to the simplest clear statements we can make, so that we can prove everything we say on the basis of those "axioms". Let's look at what they mean. Substitution: Suppose I know that "big" is just another word for "large". That means that anywhere I read the word "large", I can replace it with the word "big". That's obvious. (Though in language, sometimes things get a lot more tricky than in math, since words can have more than one meaning!) The same thing works in math. If I know that two variables both refer to the same number (that's what "equal" means), then anything I say about one of them is also true of the other. If I know a = b and you give me the expression 3a^2 - 5 (that is, 3 times the square of a, minus 5), then I know I can replace "a" with "b" in that expression, and 3b^2 - 5 will always have exactly the same value. If, for example, a and b are both equal to 4, then regardless of whether I call it a or b, I get the same answer (43) when I evaluate the expression. Symmetric property: The word "symmetry" means that if we reverse something, it still looks the same. For example, the word "wow" is symmetrical, because if I look at it in a mirror so that it is reversed, it looks the same. Equality is "symmetrical" in a similar sense: if I have an equation a = b then if I read it backwards, as b = a it is still true. It's obvious that if this is equal to that, then that is also equal to this. But we call attention to this because there are other "relations" we can define that are NOT symmetrical. For example, suppose instead of "=" we were talking about ">". Then if we take the fact 1 < 3 which is true, and turn it around to get 3 < 1 it becomes false. The "less than" relation is NOT symmetrical. But, knowing that equality IS symmetrical, we know that if we have an equation like 3x - 5 = y we can rewrite that as y = 3x - 5 and it is still true. That is important in solving equations. Transitive property: The word "transitive" means "going through"; it's related to a "transition", which "goes through" from one situation to another. Going back to words, since "big" means the same as "large", and "large" means the same as "sizeable", we know that "big" means the same as "sizeable". In algebra, we say that if we have three variables a, b, and c, and we know that a = b and b = c then the three variables are ALL equal to one another, and a = c In simple words, if two things are equal to a third thing, then they are equal to one another. As an example, if we know that 3x - 5 = 2y + 3 and that 2y + 3 = 5z - 1 then we can conclude that 3x - 5 = 5z - 1 Using specific numbers in place of the variables is tricky if you want to illustrate these properties: since everything is equal, all the numbers would have to be the same! The transitive property, for example, says that since we know that 1=1 and 1=1, therefore 1=1. Not very interesting! It works a lot better to do as I have done above and use variables or words as examples. But try this: suppose I know that my math book weigh the same as my history book, and my history book weighs the same as two English books. Then the transitive property tells me that my math book weighs the same as two English books. Does that make it clearer? If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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