there is a binary operation * defined on Z by a*b = 3a+5b . Prove that the usual multiplication on Z distributes over *
Answers
Answer:
It is not because a binary operation on a set takes two elements of that set and produces an element of that set as well.
This operation fails to do that in the case that the subtraction of two positive integers happens to be negative.
For example 2 and 5 are members of Z+. But 2*5 = 2 - 5 = -3 which is not in Z+.
Step-by-step explanation:
The following are binary operations on Z :
The arithmetic operations, addition + , subtraction − , multiplication × , and division ÷ .
Define an operation oplus on Z by a⊕b=ab+a+b,∀a,b∈Z .
Define an operation ominus on Z by a⊖b=ab+a−b,∀a,b∈Z .
Define an operation otimes on Z by a⊗b=(a+b)(a+b),∀a,b∈Z .
Define an operation oslash on Z by a⊘b=(a+b)(a−b),∀a,b∈Z .
Define an operation min on Z by a∨b=min{a,b},∀a,b∈Z .
Define an operation max on Z by a∧b=max{a,b},∀a,b∈Z .
Define an operation defect on Z by a∗3b=a+b−3,∀a,b∈Z .
Lets explore the binary operations, before we proceed:
Example 1.1.2 :
2⊕3=(2)(3)+2+3=11 .
2⊗3=(2+3)(2+3)=25 .
2⊘3=(2+3)(2−3)=−5 .
2⊖3=(2)(3)+2−3=5 .
2∨3=2 .
2∧3=3 .
Exercise 1.1.2
−2⊕3 .
−2⊗3 .
−2⊘3 .
−2⊖3 .
−2∨3 .
−2∧3 .
Answer
Properties:
Closure property
Definition : Closure property
Let S be a non-empty set. A binary operation ⋆ on S is said to be a closed binary operation on S , if a⋆b∈S,∀a,b∈S .
Below we shall give some examples of closed binary operations, that will be further explored in class.
Example 1.1.3 : Closed binary operations
The following are closed binary operations on Z .
The addition + , subtraction − , and multiplication × .
Define an operation oplus on Z by a⊕b=ab+a+b,∀a,b∈Z .
Define an operation ominus on Z by a⊖b=ab+a−b,∀a,b∈Z .
Define an operation otimes on Z by a⊗b=(a+b)(a+b),∀a,b∈Z .
Define an operation oslash on Z by a⊘b=(a+b)(a−b),∀a,b∈Z .
Define an operation min on Z by a∨b=min{a,b},∀a,b∈Z .
Define an operation max on Z by a∧b=max{a,b},∀a,b∈Z .
Define an operation defect on Z by a∗3b=a+b−3,∀a,b∈Z .
Exercise 1.1.1
Determine whether the operation ominus on Z+ is closed?
Answer
Example 1.1.4 : Counter Example
Division ( ÷ ) is not a closed binary operations on Z .
2,3∈Z but 23∉Z .
Summary of arithmetic operations and corresponding sets:
+ × − ÷
Z+ closed closed not closed not closed
Z closed closed closed not closed
Q closed closed closed closed (only when 0 is not included)
R closed closed closed closed (only when 0 is not included)