Table of Contents
What property tells that for any real number A and B if AB 0 then either a 0 or B 0?
The Zero Product Property
The Zero Product Property simply states that if ab=0 , then either a=0 or b=0 (or both).
How do you prove AB then ACBC?
Let a, b, and c be real numbers. Theorem: If a>b and b>c then a>c. Proof: Since a>b and b>c, it follows that a-b and b-c are positive real numbers (by definition of >). The sum of positive real numbers is positive, hence a-b + b-c = a-c is a positive real number.
What property is a/b is a real number?
The commutative properties tell you that two numbers can be added or multiplied in any order without affecting the result. Let a and b represent real numbers.
What property of equality is X X?
Reflexive Property
| PROPERTIES OF EQUALITY | ||
|---|---|---|
| Reflexive Property | For all real numbers x , x=x . A number equals itself. | These three properties define an equivalence relation |
| Distributive Property | For all real numbers x,y, and z , x(y+z)=xy+xz | For more, see the section on the distributive property |
What is the proof that ab = 0?
We have to show that if ab = 0, then either a = 0 or b = 0, and we have to show that if a = 0 or b = 0, then ab = 0. Let’s do the second one first because it’s easier. Proof: Given a = 0 or b = 0, to show that ab = 0. If we know that 0 times anything is 0, then we can conclude that.
What is the proof of the if and only if theorem?
As this is an “if and only if” statement, its proof requires two parts. We have to show that if ab = 0, then either a = 0 or b = 0, and we have to show that if a = 0 or b = 0, then ab = 0. Let’s do the second one first because it’s easier. This theorem is valid in any field.
How do you prove that a times B is 0?
If a is 0, then a times b is 0, but if b is 0 then b times a is 0. Q.E.D. (Note that for this half of the proof, we needed to know x0 = 0x = 0 for all x. That isn’t usually taken as an axiom for fields, but is proved from the axioms.) Proof: Given ab = 0, to show that either a = 0 or b = 0.
How to prove that a – 1 = 1?
Proof: Given ab = 0, to show that either a = 0 or b = 0. If a = 0, then we’re done, so we’ll consider the case a ≠ 0. Then by one of the axioms for a field, a has a reciprocal, a − 1, so that a − 1 a = 1.