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What are the kinds of axioms?
An Axiom is a mathematical statement that is assumed to be true. There are five basic axioms of algebra. The axioms are the reflexive axiom, symmetric axiom, transitive axiom, additive axiom and multiplicative axiom.
Describe the role of axiom systems in algebra?
An axiom in algebra is the stepping stone to solving equations. In order to solve and equation you know how to use the commutative, associative, distributive, transitive and equalilty axiom to solve the basic steps. For example: if you want an equation in the form y = mx + b, given 6x - 3y = 9 you must subtract 6x from both sides giving: -3y = 9-6x. Then you divide by -3 to get y = -3 + 2x. But the equation is not in the from y = mx + b. So we use the commutative property to switch the -3 + 2x and make it 2x - 3. Now it become y = 2x -3. and it is in the form y = mx + b. This manipulation could not be perfromed unless tahe student knew the commutative property. Once the axiom is know the algebraic manipulations fall into place.
What is the difference between the zero property of multiplication and the identity property of addition?
Usually, the identity of addition property is defined to be an axiom (which only specifies the existence of zero, not uniqueness), and the zero property of multiplication is a consequence of existence of zero, existence of an additive inverse, distributivity of multiplication over addition and associativity of addition. Proof of 0 * a = 0: 0 * a = (0 + 0) * a [additive identity] 0 * a = 0 * a + 0 * a [distributivity of multiplication over addition] 0 * a + (-(0 * a)) = (0 * a + 0 * a) + (-(0 * a)) [existence of additive inverse] 0 = (0 * a + 0 * a) + (-(0 * a)) [property of additive inverses] 0 = 0 * a + (0 * a + (-(0 * a))) [associativity of addition] 0 = 0 * a + 0 [property of additive inverses] 0 = 0 * a [additive identity] A similar proof works for a * 0 = 0 (with the other distributive law if commutativity of multiplication is not assumed).
Through a given point on a given line there is exactly one line parallel to the given line what does it define?
Playfair Axiom
What if -x plus x equals 0x?
This is always true! According to the Additive Inverse Axiom -X+X always equals 0 which is equivalent to 0X.
