Activity 1
1. State whether each of the following symbolic statements is true $(T)$ or false $(F)$. If the statement is true $(T)$, state the property/law that was applied.
For any three numbers $\otimes, \oplus$ and $\otimes$.
(a) $\otimes+\nabla=\nabla+$
(b) $\Delta+\Delta \Delta=\Delta \Delta+\Delta$
(c) $\nabla+(\otimes)+\otimes)=(\nabla+\otimes)+$
(d) $\hat{\theta} \times \vec{\nabla}=\vec{\theta}$
(e) $\nabla \times(\vec{\nabla} \times \otimes)=(\nabla \times \otimes) \times$
(f) $(\boxtimes \nabla \times \nabla) \times \otimes=\otimes \times(\nabla+\otimes)$
Answer (2)
Three statements are true based on the commutative and associative properties of addition, while the last three statements are false due to the properties of vector cross products. The findings clarify how mathematical operations interact with each other under certain rules. Understanding these properties is crucial for handling algebraic expressions and vector operations accurately.
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Statement (a) is true due to the commutative property of addition: ⊗ + ∇ = ∇ + ⊗ .
Statement (b) is true due to the commutative property of addition: Δ + ΔΔ = ΔΔ + Δ .
Statement (c) is true due to the associative property of addition: ∇ + ( ⊗ + ⊗ ) = ( ∇ + ⊗ ) + ⊗ .
Statements (d), (e), and (f) are false because they do not follow the properties of cross products and addition. a) True, b) True, c) True, d) False, e) False, f) False
Explanation
Problem Analysis We are asked to determine whether the given symbolic statements are true or false, and if true, to state the property/law that applies.
Statement (a) (a) ⊗ + ∇ = ∇ + ⊗ . This statement is true due to the commutative property of addition, which states that the order in which numbers are added does not affect the sum.
Statement (b) (b) Δ + ΔΔ = ΔΔ + Δ . This statement is also true due to the commutative property of addition.
Statement (c) (c) ∇ + ( ⊗ + ⊗ ) = ( ∇ + ⊗ ) + ⊗ . This statement is true due to the associative property of addition, which states that the way in which numbers are grouped when adding does not affect the sum.
Statement (d) (d) θ ^ × ∇ = θ . This statement is false. The cross product is anti-commutative, meaning that θ ^ × ∇ = − ( ∇ × θ ^ ) . Also, the result of a cross product is a vector, but the magnitudes are not necessarily equal.
Statement (e) (e) ∇ × ( ∇ × ⊗ ) = ( ∇ × ⊗ ) × ⊗ . This statement is false. The cross product is not associative, meaning that the order of operations matters. In general, a × ( b × c ) = ( a × b ) × c .
Statement (f) (f) ( ⊠ ∇ × ∇ ) × ⊗ = ⊗ × ( ∇ + ⊗ ) . This statement is false. The left side involves a cross product and the right side involves a cross product and addition. There is no general property that makes this true.
Final Answer Therefore, the answers are: (a) True, Commutative Property of Addition (b) True, Commutative Property of Addition (c) True, Associative Property of Addition (d) False (e) False (f) False
Examples
The commutative and associative properties are fundamental in many areas of mathematics and physics. For example, when calculating the total resistance in a series circuit, the order in which you add the individual resistances doesn't matter (commutative property). Similarly, when calculating the total energy of a system, the way you group the energy contributions doesn't affect the final result (associative property). These properties simplify calculations and ensure consistency in results.
