Important Note
The homework should be written based on your understanding but working with someone is fine.Attendace List
Date: October 3, 2021
- Daniel
- Lucas
- John
- George
Agenda
- Review lecture notes and definitions from book
- Go over exercises from the book
- Clarify any question and doubts
Uncovered material from last week
What is a binary operator?
"A binary operation * on a set S is a function mapping S x S into
S.
For each (a, b) E S x S, we will denote the element *((a, b)) of S by a* b.
Intuitively, we may regard a binary operation * on S as assigning, to each ordered pair (a, b)
ofelements of S, an element a * b of S.
Binary refers to the fact that we are mapping pairs of elements from S into S."(Section1, Pg 11)
Examples:
Note: Suppose we wish to consider an expression of the form a * b * c. A binary operation * enables us to combine only two elements, and here we have three.
- "Our usual addition + is an operation on the set R" (Section1, Pg 11)
- "Let M(R) be the set of all matricest with real entries. The usual matrix addition + is not an operation on this set since A + B is not defined for an ordered pair (A, B) of matrices having different numbers of rows or of columns."(Section1, Pg 12)
New material
What is an identity element>
"Let S be a set with binary operation *· If e E S has the property that for all a E S,
a * e = e *a = a, then e is called an identity element for
*·"(Section1, Pg 14)
What is an inverse?
"If* is an operation on the set S and S has an identity element e, then for any x ES, the inverseof x is an element x' such that x * x' = x' * x =
e."(Section1, Pg
13)
When is * consider commuative?
"An operation * on a set S is commutative if (and only if) a * b = b *a for all
a,b in S."(Section1, Pg 13)
Details
As was pointed out in Section 0, it is customary in mathematics to omit the words and only if from a definition. Definitions are always understood to be if and only if statements. Theorems are not always if and only if statements, and no such convention is ever used for theorems.Examples:
- The indetity element for addition is 0 i.e a + 0 = a where a is an element of real numbers.
- The inverse for addition is usually the opposite of a given number i.e a + (-a) = 0 where is an element of real numbers.
Questions and Concerns
- When are homework assignments due?
- Assignments are due on Monday's by 11:59pm.
- How are we turning in the homework?
- Homework assignmnets should be turned in person, both typed and handwritten is accepet.
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