State whether each of the following mathematical statements is true $(T)$ or false $(F)$. If the statement is true, state the law that was applied.
(a) $5+4=5 \times 4$
(b) $7+5=5+7$
(c) $3+(8+6)=(3+8)+6$
(d) $4+7+2=(4+7)+2$
(e) $7 \times 4=4 \times 7$
(f) $2 \times 6=6 \div 2$
(g) $2 \times(3 \times 8)=(2 \times 3) \times 8$
(h) $5 \times(3 \times 4)=5 \times(4+3)$
Answer (1)
Evaluate 5 + 4 and 5 × 4 and conclude that 5 + 4 = 5 × 4 is false.
Evaluate 7 + 5 and 5 + 7 and conclude that 7 + 5 = 5 + 7 is true due to the commutative property of addition.
Evaluate 3 + ( 8 + 6 ) and ( 3 + 8 ) + 6 and conclude that 3 + ( 8 + 6 ) = ( 3 + 8 ) + 6 is true due to the associative property of addition.
Evaluate 4 + 7 + 2 and ( 4 + 7 ) + 2 and conclude that 4 + 7 + 2 = ( 4 + 7 ) + 2 is true due to the associative property of addition.
Evaluate 7 × 4 and 4 × 7 and conclude that 7 × 4 = 4 × 7 is true due to the commutative property of multiplication.
Evaluate 2 × 6 and 6 \tdiv 2 and conclude that 2 × 6 = 6 \tdiv 2 is false.
Evaluate 2 × ( 3 × 8 ) and ( 2 × 3 ) × 8 and conclude that 2 × ( 3 × 8 ) = ( 2 × 3 ) × 8 is true due to the associative property of multiplication.
Evaluate 5 × ( 3 × 4 ) and 5 × ( 4 + 3 ) and conclude that 5 × ( 3 × 4 ) = 5 × ( 4 + 3 ) is false.
The final answers are: (a) False (b) True (c) True (d) True (e) True (f) False (g) True (h) False
Explanation
Analyzing the Statements Let's analyze each statement to determine if it's true or false and, if true, identify the applied mathematical law.
(a) 5 + 4 = 5 × 4
LHS: 5 + 4 = 9
RHS: 5 × 4 = 20
Since 9 e q 20 , the statement is false.
(b) 7 + 5 = 5 + 7
LHS: 7 + 5 = 12
RHS: 5 + 7 = 12
Since 12 = 12 , the statement is true. This illustrates the commutative property of addition.
(c) 3 + ( 8 + 6 ) = ( 3 + 8 ) + 6
LHS: 3 + ( 8 + 6 ) = 3 + 14 = 17
RHS: ( 3 + 8 ) + 6 = 11 + 6 = 17
Since 17 = 17 , the statement is true. This illustrates the associative property of addition.
(d) 4 + 7 + 2 = ( 4 + 7 ) + 2
LHS: 4 + 7 + 2 = 11 + 2 = 13
RHS: ( 4 + 7 ) + 2 = 11 + 2 = 13
Since 13 = 13 , the statement is true. This illustrates the associative property of addition.
(e) 7 × 4 = 4 × 7
LHS: 7 × 4 = 28
RHS: 4 × 7 = 28
Since 28 = 28 , the statement is true. This illustrates the commutative property of multiplication.
(f) 2 × 6 = 6 \tdiv 2
LHS: 2 × 6 = 12
RHS: 6 \tdiv 2 = 3
Since 12 e q 3 , the statement is false.
(g) 2 × ( 3 × 8 ) = ( 2 × 3 ) × 8
LHS: 2 × ( 3 × 8 ) = 2 × 24 = 48
RHS: ( 2 × 3 ) × 8 = 6 × 8 = 48
Since 48 = 48 , the statement is true. This illustrates the associative property of multiplication.
(h) 5 × ( 3 × 4 ) = 5 × ( 4 + 3 )
LHS: 5 × ( 3 × 4 ) = 5 × 12 = 60
RHS: 5 × ( 4 + 3 ) = 5 × 7 = 35
Since 60 e q 35 , the statement is false.
Final Answers Here's the summary of our analysis:
(a) 5 + 4 = 5 × 4 - False (b) 7 + 5 = 5 + 7 - True (Commutative Property of Addition) (c) 3 + ( 8 + 6 ) = ( 3 + 8 ) + 6 - True (Associative Property of Addition) (d) 4 + 7 + 2 = ( 4 + 7 ) + 2 - True (Associative Property of Addition) (e) 7 × 4 = 4 × 7 - True (Commutative Property of Multiplication) (f) 2 × 6 = 6 \tdiv 2 - False (g) 2 × ( 3 × 8 ) = ( 2 × 3 ) × 8 - True (Associative Property of Multiplication) (h) 5 × ( 3 × 4 ) = 5 × ( 4 + 3 ) - False
Examples
Understanding these properties is crucial in various fields. For instance, in computer science, the associative property is used in parallel computing to break down complex calculations into smaller, independent tasks that can be executed simultaneously. The commutative property simplifies calculations in physics, such as when calculating the net force acting on an object, where the order of adding force vectors does not affect the result. These basic mathematical principles underpin many advanced concepts and practical applications.
