Theorem M1: If x is a number, y is a nonzero number, and xy=y, then x=1. Theorem

Question

Theorem M1: If x is a number, y is a nonzero number, and xy=y, then x=1.

Theorem M2: If x is a nonzero number, y is a number, and xy=1, then y=(/x)

Theorem M3: If each of x, y, and z is a number, y?0, and yz=x, then z=x(/y)

Theorem M4: If x is a nonzero number, then /(/x)=x

Theorem M5: /1=1

Usable Axioms (use necessary axioms to prove theorems above):

Axiom A1: A list of numbers may be added in any order, with any grouping, without changing the resulting sum.

Axiom A2: There is a number called zero, denoted by 0, such that if 'a' is a number then 'a'+0='a'.

Axiom A3: If 'a' is a number then -'a'(read minus a) is a number and 'a'+(-a)=0.

Axiom M1: A list of numbers may be multiplied in any order, with any grouping, without changing the resulting product.

Axiom M2: There is a number called one, denoted by 1, such that if 'a' is a number, then 1a=a

Axiom M3: If 'a' is a nonzero number, then /a (read slash 'a') is a number and 'a'(/a)=1

Axiom D: If each of x, y, and z is a number then x(y+z)=xy+xz

Axiom NE: 0 does not equal 1

Explanation / Answer

according to axiom M2 : 1a=a 1y=y and xy=y comparing two equations x=1 xy=1, multiply by 1/x x(1/x)y=1(1/x) from M3 we can say y=(1/x) yz=x multiply by 1/y y(1/y)z=x(1/y) from M3 we can say z=x(1/y) x(1/x) =1 from M3 dividing both sides by (1/x) we get 1/(1/x) = x(1/x)/(1/x) from M3 1/(1/x) = x similarly put x=1 1/1/1 =1

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