Flashcards
Binary Operations and Modular Arithmetic
Flashcards do conjunto (26)
- 1What is a binary operation?It takes 2 elements from a set and performs an operation on them
- 2What does "∈" mean?Element of...
- 3In a binary operation, we can use any.....Any symbol
(e.g: a ⚪ b = a + b for all a,b ∈ R)
(e.g: a Ø b = a^2 + b^3 for all a,b ∈ N) - 4What does a ⚪ b = a + b for all a,b ∈ R do?Adds any 2 elements from the set of all real numbers
- 5What does a Ø b = a^2 + b^3 for all a,b ∈ N do?Finds sum of first element squared and second element cubed where the elements are from the set of all natural numbers
- 6What is the notation for modular arithmetic?a=b (mod n) where a,b,n are integers
- 7Using a=b (mod n), what will a/n and b/n both give?The same remainder
- 8Mod is short for.....Modulo
- 9Mod 12 represents....A group of 12 numbers where addition and subtraction loops around in a group of 12 (like a clock; 11+2 = 1)
- 10In (mod 10), 83 and 3 both are.....The same number
- 11If we have (mod 5) , what are the possible remainder values?0,1,2,3,4
- 12a+b(mod n) can also be written like....a+(n)b
Note that (n) is small
This is the same for a x b(mod n) = a x (n)b - 13Using a=b(mod n) means that a=pn+r and b=qn+r . Prove this.a+(n)b = (pn+r) + (n)(qn+r)
pn+(n)qn+(n)r+(n)r
(n)(p+q) + 2r(mod n)
(n)(p+q) must = 0 , so 2r(mod n)
(same for b) - 14Using a=b(mod n) means that a=pn+r and b=qn+r . Prove this (using a x (n)n)a x (n)n = (pn+r) x (n)(qn+r)
pqn^2 + pnr + qnr + r^2 (mod n)
n(pqn+pr+qr) + r^2 (mod n)
Becomes r^2 (mod n) as n(pqn+pr+qr) is cancelled due to modular arithmetic laws (must = 0) - 15If the output of a binary operation is from the same set as the input(s) then the operation is said to be.....Closed
(e.g. addition and multiplication from real positive integers) - 16A binary operation is commutative if....a⌾b = b⌾a (a,b ∈ S)
- 17A binary operation is associative if....(a◊b)◊c = a◊(b◊c) (a,b,c ∈ S)
- 18If a binary operation has an identity then....e ◊ x = x and x ◊ e = x (e,x ∈ S)
(e is the identity element and is unique to the binary operation)
(Note not all binary operations have an identity) - 19Is x*y = x^2 + y^2 + 2xy commutative? (For all naturals)y*x = y^2 + x^2 + 2yx
Same as x*y so yes it's commutative - 20Is x◾y = 1/x - 1/y + 1/xy commutative? (For all integers)x◾y = 1/x - 1/y + 1/xy same as :
x-y+1 / xy
y◾x = 1/y - 1/x + 1/yx same as :
y-x+1 / xy
Use counter example to prove not commutative (e.g. x=2 , y=3) - 21a⊠b = 2a+b for all reals. Is it associative?(a⊠b)⊠c = (2a+b)⊠c = 2(2a+b)+c = 4a+2b+c
a⊠(b⊠c) = a⊠(2b+c) = 2a+2b+c
Use counter example to prove not associative (e.g. a=1, b=2 , c=3) to show they arent equal - 22A◊B = AB where A,B are 2x2 matrices. Are they associative?(A◊B)◊C = (AB)C
A◊(B◊C) = A(BC)
This shows they are associative and we know matrices are associative anyway - 23Find the identity element for x*y = 2xy for all naturalse*x = x , 2ex=x , 2e=1, e=1/2
x*e=x , 2xe = x, 2x=1, x=1/2
Verify: 5*1/2 ; 2x5x1/2 = 5 so identity element is 1/2 - 24Find the identity element for x⊡y = 2x+2y for all naturalse⊡x = x , 2e+2x=x, 2e+x=0, x=-2e, e=-x/2
x⊡e=x, 2x+2e=x, x+2e=0, x=-2e , e=-x/2
(no identity as there is an unknown in both sides) - 25The identity element for addition is always....0
- 26The identity element for multiplication is always....1
Informações
- Descrição
- Binary Operations and Modular Arithmetic
- Criado por
- Doc Boff
- Visualizações
- 4
- Avaliação