## Problem 298

Problem 298

Study the following diagram:

i. Characteristics of addition that are different from multiplication

ii. Characteristics that are common to both operations

iii. Characteristics of multiplication that are different from addition

a. Complete the diagram to illustrate how the characteristics of multiplication and addition of whole numbers compare.

b. Prepare a similar diagram for subtraction and division.

Solution

a.

i.                     The identity symbol for addition is the zero and identity symbol for multiplication is one.

ii.                   Multiplication and addition are associative, commutative, closure and have an identity property

iii.                 Multiplication has a distributive and zero property, addition does not.

b.

## Problem 297

Problem 297

Use the distributive property of multiplication over addition to find each product:

a. (20 + 5) * 3

b. 4 * (5 + 6)

C. (x + 10) * (3x + 2)

Solution

a. (20 + 5) * 3 = (20 * 3) + (5 * 3) = 60 + 15 = 75

b. 4*(5 + 6) = (4 * 5) + (4 * 6) = 20 + 24 = 44

c. (x + 10) * (3x + 2) = (x)(3x) + (x)(2) + (10)(3x) + (10)(2) = 3x2 + 2x + 30x + 20  = 3x2 + 32x + 20

## Problem 296

Problem 296

Properties of addition may be investigated in relation to sets other then the set of whole numbers. Use the set of multiples of 3, {0 ,3,6,9,….} to answer the following questions, and then justify answers:

a. Is this set closed under addition?

b. Does this set have an additive identity element?

c. Is addition with this set commutative?

d. Is addition with this set associative?

Solution

a. yes

b. yes

c. yes

d. yes

## Problem 295

Problem 295

Use the definition of subtraction to determine the subtraction equation that are related to each of the following addition equations:

a. 8 + 7 = n

b. 14 + x = 25

c. r + s = t

Solution

a. 8 = n-7

b.  x = 25-14

c.  r = t-s

## Problem 294

Problem 294

Use sets to verify the following answers:

a. 4 + 5 = 9

b. 0 + 6 = 6

c. 10 + 2 = 12

Solution

a. {11,12,13,14}+{15,16,17,18,19} = {11,12,13,14,15,16,17,18,19}

b. {}+{a,b,c,d,e,f} = {a,b,c,d,e,f}

c. {1,2,3,4,5,6,7,8,9,10}+{11,12} ={1,2,3,4,5,6,7,8,9,10,11,12}

## Problem 293

Problem 293

Decide whether the following statements are true or false and justify your answer:

a. If n(A) < n(B), then A B.

b. If n(A) n(B) , then A   B.

c. If A   B,  then n(A)< n(B).

d. If A B, then n(A) n(B).

Solution

a. False, elements are not same in both sets.

b. False , not equal or equivalent

c. True , equal or equivalent

d. True , equal or equivalent

## Problem 292

Problem 292

a. If two sets are equal, are they necessarily equivalent?

Why or why not?

b. If two sets are equivalent, are they necessarily equal?

Why or why not?

c. Draw a diagram of and write a statement describing the relationship between the ideas of two sets being equivalent and the idea of two sets being equal.

Solution

a. Yes, equal means both have same element and all equal sets are always equivalent also.

b. NO, equivalent shows number of elements are same may be they are same but not necessarily.

c. Equal sets have same elements and same number of elements

Equivalent sets have same number of elements but not same elements.

Equal    A={1,2,3} , B = {1,2,3}

Equivalent  A= {1,2,3}, B= {A,B,C}

## Problem 291

Problem 291

How are the ideas of subsets and proper subsets used in counting to identify relationships between whole numbers?

Solution

It tell us which is less and which is greater than between the whole numbers. If a proper set is equivalent to a other set it means that other set is less than the first one.

## Problem 290

Problem 290

Use sets to verify that 8 > 6.

Solution

{a ,b ,c ,d ,e ,f ,g ,h} > {a ,b ,c ,d ,e ,f}

Now we can see first set has 2 additional elements, so 8 > 6

## Problem 289

Problem 289

Use the following sets to complete the given statements in as many ways as possible:

A = {a, b, c, d}

B = {e , f}

C = {b ,a , c, d}

D = {1 , 2 , 3, 4}

a. Sets _____and____are not equivalent and not equal.

b. Sets_____and____are not equivalent but are equal.

c. Sets_____and ____are equivalent but not equal.

d. Sets_____and____are equivalent and  equal.

Solution

a.      A and B , B and C , B and D

b.      No any combination.

c.        A and D, C and D

d.      A and C