Examkin

Soru

Zorluk: OrtaFunctions and Custom Symbol Operations

For all real numbers xx and yy, the custom binary operation ⊙\odot is defined by x⊙y=x+y−xyx \odot y = x + y - xy. Which of the following statements must be true for all real numbers aa, bb, and cc? Select all such statements.

  1. a⊙b=b⊙aa \odot b = b \odot aCevap
  2. a⊙1=1a \odot 1 = 1Cevap
  3. C
    a⊙0=0a \odot 0 = 0
  4. (a⊙b)⊙c=a⊙(b⊙c)(a \odot b) \odot c = a \odot (b \odot c)Cevap
  5. E
    a⊙a=a2a \odot a = a^2

Cevap

The statements a⊙b=b⊙aa \odot b = b \odot a, a⊙1=1a \odot 1 = 1, and (a⊙b)⊙c=a⊙(b⊙c)(a \odot b) \odot c = a \odot (b \odot c) must be true for all real numbers aa, bb, and cc.
The custom operation is commutative (a⊙b=b⊙aa \odot b = b \odot a), maps any real number operated with 11 to 11 (a⊙1=1a \odot 1 = 1), and is associative ((a⊙b)⊙c=a⊙(b⊙c)(a \odot b) \odot c = a \odot (b \odot c)). These three properties hold universally for all real numbers.

Adım Adım Çözüm

1
Evaluate a⊙ba \odot b and b⊙ab \odot a to test commutativity.
a⊙b=a+b−aba \odot b = a + b - ab and b⊙a=b+a−bab \odot a = b + a - ba. Since addition and multiplication of real numbers are commutative, a+b−ab=b+a−baa + b - ab = b + a - ba.
Verify if order of operands alters the result.
2
Evaluate a⊙1a \odot 1 using the operation definition.
a⊙1=a+1−a(1)=a+1−a=1a \odot 1 = a + 1 - a(1) = a + 1 - a = 1.
Test the behavior of operating with the constant 11.
3
Evaluate a⊙0a \odot 0 to verify the zero property statement.
a⊙0=a+0−a(0)=aa \odot 0 = a + 0 - a(0) = a. This equals aa, not 00 for general values of aa.
Check if operating with 00 results in 00.
4
Evaluate both sides of (a⊙b)⊙c=a⊙(b⊙c)(a \odot b) \odot c = a \odot (b \odot c) to test associativity.
Left side: (a⊙b)⊙c=(a+b−ab)⊙c=(a+b−ab)+c−(a+b−ab)c=a+b+c−ab−ac−bc+abc(a \odot b) \odot c = (a + b - ab) \odot c = (a + b - ab) + c - (a + b - ab)c = a + b + c - ab - ac - bc + abc. Right side: a⊙(b⊙c)=a⊙(b+c−bc)=a+(b+c−bc)−a(b+c−bc)=a+b+c−bc−ab−ac+abca \odot (b \odot c) = a \odot (b + c - bc) = a + (b + c - bc) - a(b + c - bc) = a + b + c - bc - ab - ac + abc. Both sides are identical.
Verify whether grouping alters the result.
5
Evaluate a⊙aa \odot a.
a⊙a=a+a−a(a)=2a−a2a \odot a = a + a - a(a) = 2a - a^2. This is not equal to a2a^2 except when 2a−a2=a2  ⟹  2a2−2a=0  ⟹  a=02a - a^2 = a^2 \implies 2a^2 - 2a = 0 \implies a=0 or a=1a=1.
Determine if the self-operation yields a2a^2 for all real numbers.

Anahtar Kavram

Evaluating algebraic properties (commutativity, identity, associativity) of custom binary operations.
Bu soruyu puanla

Konular

Functions and Custom Symbol Operations

Bu konudan daha fazla soru