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4 years ago

Match the solution set given in inequality notation with the solution set given in interval notation.

Mathematics

1 answer:

iragen [17]4 years ago

6 0

x ≤ a → x ∈ (-∞, a]

x < a → x ∈ (-∞, a)

x ≥ a → x ∈ [a, ∞)

x < a → x ∈ (a, ∞)

--------------------------------------

Therefore:

x ≥ 7.8 → [7.8, ∞)

x < 7.8 → (-∞, 7.8)

x ≤ 7.8 → (-∞, 7.8]

x > 7.8 → (7.8, ∞)

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Let A, B, C and D be sets. Prove that A \ B and C \ D are disjoint if and only if A ∩ C ⊆ B ∪ D

ANEK [815]

Step-by-step explanation:

We have to prove both implications of the affirmation.

1) Let's assume that A \ B and C \ D are disjoint, we have to prove that A ∩ C ⊆ B ∪ D.

We'll prove it by reducing to absurd.

Let's suppose that A ∩ C ⊄ B ∪ D. That means that there is an element x that belongs to A ∩ C but not to B ∪ D.

As x belongs to A ∩ C, x ∈ A and x ∈ C.

As x doesn't belong to B ∪ D, x ∉ B and x ∉ D.

With this, we can say that x ∈ A \ B and x ∈ C \ D.

Therefore, x ∈ (A \ B) ∩ (C \ D), absurd!

It's absurd because we were assuming that A \ B and C \ D were disjoint, therefore their intersection must be empty.

The absurd came from assuming that A ∩ C ⊄ B ∪ D.

That proves that A ∩ C ⊆ B ∪ D.

2) Let's assume that A ∩ C ⊆ B ∪ D, we have to prove that A \ B and C \ D are disjoint (i.e. A \ B ∩ C \ D is empty)

We'll prove it again by reducing to absurd.

Let's suppose that A \ B ∩ C \ D is not empty. That means there is an element x that belongs to A \ B ∩ C \ D. Therefore, x ∈ A \ B and x ∈ C \ D.

As x ∈ A \ B, x belongs to A but x doesn't belong to B.

As x ∈ C \ D, x belongs to C but x doesn't belong to D.

With this, we can say that x ∈ A ∩ C and x ∉ B ∪ D.

So, there is an element that belongs to A ∩ C but not to B∪D, absurd!

It's absurd because we were assuming that A ∩ C ⊆ B ∪ D, therefore every element of A ∩ C must belong to B ∪ D.

The absurd came from assuming that A \ B ∩ C \ D is not empty.

That proves that A \ B ∩ C \ D is empty, i.e. A \ B and C \ D are disjoint.

7 0

3 years ago

Kaya used these steps to solve the equation 7 + 2x = 2(x - 1) + 4. Which choice describes the meaning of her result, 7 = 2?

Whitepunk [10]

Answer is A

7 + 2X = 2(X-1) + 4

7 + 2X = 2X-2+4

7 + 2X = 2X + 2 whether 2X = 2X-5 or 5 + 2X = 2X still 0 = 5 Or 0= -5 which is not possible

4 0

4 years ago

Read 2 more answers

Write the ratio in simplest form: 27 red markers to 12 blue markers

elena-14-01-66 [18.8K]

What do you mean by this? Do you have a picture

6 0

3 years ago

Write the product in its simplest form:<br> 7xy3 . (-9xy)

Naddik [55]

The answer would be
-63x^2y^3

3 0

3 years ago

1. What is the median of the data set?

Tamiku [17]

Answer: 1. 7

2. 100

3. 193.8

4. 0.8

5. 90.6

Step-by-step explanation:

1. Given the data 9,3,10,12,4,5,12,2

For finding the median we arrange it in ascending order

2,3,4,5,9, 10,12,12

Since the no of observations are even

∴ median = $\frac{n/2 th term + (n/2 + 1)th term}{2}$

= $\frac{5+9}{2}$

= $\frac{14}{2}$

i.e. Median =7

2. Given the data

23, 95,100,23,100,100

Since the no 100 is repeated thrice i.e. the maximum no of times

∴ Mode=100

Given the data

108, 305,252,113, 191

Mean= $\frac{108+305+252+113+191}{5}$

= $\frac{969}{5}$

=193.8

4. Money spent in the first week

= $11.52, $6.48, $5.99, $14.00, and $9.50

Total for the first week

=11.52+6.48+5.99+14.00+9.50

=$47.49

Now she spent $4 more in the second week

i.e.$47.49+4

=51.49$

Increase in mean = $\frac{4}{5}$

=0.8

Given

2 students scored 100 each

9 students scored 95 each

10 students scored 90 each

3 students scored 80 each

1 student scored 70

Total score of students = $2\times100+ 9\times 95+10\times90+3\times80+70\times1$

=2265

The average score

= $\frac{2265}{25}$

=90.6

8 0

3 years ago