G Consider the following statement. For all sets A and B, (A − B) ∪ (A ∩ B) = A. Construct an algebraic proof for the statement. Cite a property from Theorem 6.2.2 for every step.

(A − B) ∪ (A ∩ B) = A

Step-by-step explanation:

we will use below conditions

Let A and B be the two sets

Distributive property:-

i) A U (B ∩ C) =( AUB)∩(AUC)

ii) A ∩ (BUC) = (A ∩ B) U (A ∩ C)

we will use another conditions also iii)  A-B =A ∩ B¹

now Given (A − B) ∪ (A ∩ B) = (A ∩ B¹) ∪ (A ∩ B) ( from (iii)

                                             = A ∩ (BUB¹)   ( from (ii)

                                             = A ∩ U ( see the diagram)

                                             = A

∴ (A − B) ∪ (A ∩ B) = A

g = (2, 2)

step-by-step explanation:

given the ratio is 1 : 1 then f is the midpoint of eg

let the coordinates of g = (x, y)

using the midpoint formula

(0 + x) = 1 ( multiply both sides by 2 )

0 + x = 2 ⇒ x = 2

similarly

(4 + y) = 3 ( multiply both sides by 2 )

4 + y = 6 ( subtract 4 from both sides )

y = 2

coordinates of g = (2, 2)

step-by-step explanation:

given that there are two triangles j'k'l' and jkl>

the former is got by dilating the second one.

when drawn on a grid, (graph paper) coordinates of vertices of first triangle is

(6,12)j, (9,6)k and (12,9)l

side jk has length = distance between j and k

=

side j'k'=distance between j' and k'

=

ie j'k'l' has sides 1/3 of that of corresponding sides of jkl.

hence scale factor of 1/3 is used to get j'k'l'from jkl