1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
UkoKoshka [18]
3 years ago
5

26. Define a relation ∼ ∼ on R 2 R2 by stating that ( a , b ) ∼ ( c , d ) (a,b)∼(c,d) if and only if a 2 + b 2 ≤ c 2 + d 2 . a2+

b2≤c2+d2. Show that ∼ ∼ is reflexive and transitive but not symmetric.
Mathematics
1 answer:
Tresset [83]3 years ago
3 0

Answer:

~ is reflexive.

~ is asymmetric.

~ is transitive.

Step-by-step explanation:

~ is reflexive:

i.e., to prove $ \forall (a, b) \in \mathbb{R}^2 $, $ (a, b) R(a, b) $.

That is, every element in the domain is related to itself.

The given relation is $\sim: (a,b) \sim (c, d) \iff a^2 + b^2 \leq c^2 + d^2$

Reflexive:

$ (a, b) \sim (a, b) $ since $ a^2 + b^2 = a^2 + b^2 $

This is true for any pair of numbers in $ \mathbb{R}^2 $. So, $ \sim $ is reflexive.

Symmetry:

$ \sim $ is symmetry iff whenever $ (a, b) \sim (c, d) $ then $  (c, d) \sim (a, b) $.

Consider the following counter - example.

Let (a, b) = (2, 3) and (c, d) = (6, 3)

$ a^2 + b^2 = 2^2 + 3^2 = 4 + 9 = 13 $

$ c^2 + d^2 = 6^2 + 3^2 = 36 + 9 = 42 $

Hence, $ (a, b) \sim (c, d) $ since $ a^2 + b^2 \leq c^2 + d^2 $

Note that $ c^2 + d^2 \nleq a^2 + b^2 $

Hence, the given relation is not symmetric.

Transitive:

$ \sim $ is transitive iff whenever $ (a, b) \sim (c, d) \hspace{2mm} \& \hspace{2mm} (c, d) \sim (e, f) $ then $ (a, b) \sim (e, f) $

To prove transitivity let us assume $ (a, b) \sim (c, d) $ and $ (c, d) \sim (e, f) $.

We have to show $ (a, b) \sim (e, f) $

Since $ (a, b) \sim (c, d) $ we have: $ a^2 + b^2 \leq c^2 + d^2 $

Since $ (c, d) \sim (e, f) $ we have: $ c^2 + d^2 \leq e^2 + f^2 $

Combining both the inequalities we get:

$ a^2 + b^2 \leq c^2 + d^2 \leq e^2 + f^2 $

Therefore, we get:  $ a^2 + b^2 \leq e^2 + f^2 $

Therefore, $ \sim $ is transitive.

Hence, proved.

You might be interested in
H=-16t^2+368
elixir [45]
The vertex and the t-intercept where t>0, will help solve this problem. Sending the work on the attachment.

5 0
3 years ago
I need help asap :):):)
KiRa [710]

Answer:

24 ÷ 4

Step-by-step explanation:

Putting something into fourths is putting into 4 parts. If you divide 24 by 4 you get 6. 6 equals 1/4 of 24. To check your work add 6 together 4 times.

7 0
3 years ago
Read 2 more answers
Prime factorisation of 500​
Andrews [41]

\boxed{\underline{\bf \: ANSWER}}

2 <u>| 500</u>

2 <u>| 250</u>

5 <u>| 125</u>

5 <u>| 25</u>

5 <u>| 5</u>

1 <u>| 1</u>

Prime factorisation of <u>500 = 2 × 2 × 5 × 5 × 5 × 1</u>

_____

Hope it helps.

ᏒᎯᎨᏁᏰᎾᏯᏕᎯᏝᎿ2222

6 0
2 years ago
If 2x+y=8 and x-1=y then what is y? Please be detailed I want to know more then just the answer but how to solve
IgorLugansk [536]
This is Bridge Algebra 2 stuff. Well you see that 2x+y=8 and x-1=y. To slove for y you have to slove for x first to find y. So 2x+x-1=8 ( the reason I did that is because there is a y in that equation and I placed it on that equation) now lets slove this equation for x. So 3x-1=8( I add 2x with x) 3x=9 ( I made -1 a positive one and add it to 8) x=3 ( I divided 9/3 to get 3). Now that you have slove x we will use x to slove for y. So 3-1=y, 2=y. Your answer is y=2.
4 0
3 years ago
A<br> 68°<br> B<br> 32°<br> D<br> 77<br> 11x - 33<br> [?] с<br> Help Resour<br> Skip
kari74 [83]

Quadrilaterals are shapes with 4 sides. The correct inequality expression is x < 11

<h3>What are quadrilaterals?</h3>

Quadrilaterals are shapes with 4 sides. The given figure is a quadrilateral. The side opposite to 68 degrees is greater than the side opposite 32 degrees to have the equation;

11x - 33 < 77

Add 33 to both sides

11x - 33 + 33 < 77 + 33

11x <110

x < 110/10

x< 11

Hence the correct inequality expression is x < 11

Learn more on inequality here: brainly.com/question/24372553

#SPJ1

5 0
2 years ago
Other questions:
  • Are the stars in the sky a point, line, line segment, ray, or plane
    15·1 answer
  • All integers are whole numbers?true or false​
    14·1 answer
  • Can 5/12 be simlified
    9·2 answers
  • A given line has the equation 10x 2y -2.What is the equation, in slope-intercept form, of the line that is parallel to the given
    12·1 answer
  • PLZ HELP ME!!
    10·1 answer
  • Help please I don’t understand
    9·1 answer
  • ** I need help for this one***
    14·2 answers
  • A bag contains 10 black, 15 white, and 25 blue marbles. What is the probability of randomly selecting one
    12·1 answer
  • Identify the ordered pair and quadrant (or axis) for each point on the graph.
    9·1 answer
  • Can somebody plz help answer all the questions correctly thanks !
    13·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!