Show that for any sets A and B, A (B-A)=(A )

Question

Show that for any sets A and B,

$\text{A}\cup(\text{B}-\text{A})=(\text{A}\cup\text{B})$

✓

Answer

Let $\text{x}\in\text{A}\cup\text{(B - A)}$

$\Rightarrow\text{x}\in\text{A or x}\in(\text{B - A})$

$\Rightarrow\text{x}\in\text{A or x}\in\text{B and x}\not\in\text{A}$

$\Rightarrow\text{x}\in\text{A or x}\in\text{B}$

$\Rightarrow\text{x}\in\text{(A}\cup\text{B})$

$\therefore\text{A}\cup\text{(B - A)}\subset\text{(A}\cup\text{B}).....\text{(i)}$

Let and $\text{x}\in\text{(A}\cup\text{B})$

$\Rightarrow\text{x}\in\text{A or x}\in\text{B}$

$\Rightarrow\text{x}\in\text{A or x}\in\text{B and x}\not\in\text{A}$

$\Rightarrow\text{x}\in\text{A or x}\in\text{(B - A)}$

$\Rightarrow\text{x}\in\text{A}\cup\text{(B - A)}$

$\therefore(\text{A}\cup\text{B})\subset\text{A}\cup\text{(B - A)}.....\text{(ii)}$

From (i) and (ii), we get

$\text{A}\cup\text{(B - A)}=\text{A}\cup\text{B.}$

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

	Column C₁		Column C₂
a.	In xy-plane.	i.	I^st octant.
b.	Point (2, 3, 4) lies in the.	ii.	yz-plane.
c.	Locus of the points having x coordinate 0 is.	iii.	z-coordinate is zero.
d.	A line is parallel to x-axis if and only.	iv.	z-axis.
e.	If x = 0, y = 0 taken together will represent the.	v.	plane parallel to xy-plane.
f.	z = c represent the plane.	vi.	if all the points on the line have equal y and z-coordinates.
g.	Planes x = a, y = b represent the line.	vii.	from the point on the respective.
h.	Coordinates of a point are the distances from the origin to the feet of perpendiculars.	viii.	parallel to z-axis.
i.	A ball is the solid region in the space enclosed by a.	ix	disc.
j.	Region in the plane enclosed by a circle is known as a.	x.	sphere.