Use a Venn diagram to determine which relationship, , ⊆,=, ⊇, is true for the pair of sets.
Set 1: (A − C) − (B − C)
Set 2: A − B
Let C = {2,4,6}
A = {1,2,3,4,5,6,7,9,11,13}
B = {2,4,6,9,10,11,15,16}
A-C = 1,3,5,7,9,11,13
B-C = 9,10,11,15,16
(A-C)-(B-C) = {1,3,5,7,13}
A-B = {1,3,5,7,13}
so, "="
makes sense, since A-C - (B-C) = A-C-B+C = A-B
google can provide other examples.
Ah, Venn diagrams! The "fun with circles" activity for mathematicians.
To determine the relationship between Set 1 and Set 2, let's first break down the sets:
Set 1: (A − C) − (B − C)
Set 2: A − B
Now, let's create our Venn diagram with three circles: A, B, and C.
In Set 1, (A − C) represents the elements that are in A but not in C. Then, (B − C) subtracts the elements in C from B.
In Set 2, A − B represents the elements that are in A but not in B.
To compare the two sets, we need to consider the elements in Set 1 that are not in Set 2.
If Set 1 has any elements not in Set 2, we can say that Set 1 is a superset of Set 2 (⊇). In other words, Set 2 is entirely contained within Set 1.
If Set 1 and Set 2 have exactly the same elements, then Set 1 and Set 2 are equal (=). They are essentially the same!
However, if Set 1 does not have any elements that are not in Set 2, we can say that Set 2 is a superset of Set 1 (⊆). In other words, Set 1 is entirely contained within Set 2.
But hey, let's make it a little more fun! Imagine Set 1 as a pizza and Set 2 as a slice of that pizza. If the slice is smaller and fits entirely inside the pizza, you have Set 2 ⊆ Set 1. If the slice is the same size as the pizza, it's like a perfectly cut piece, and you have Set 1 = Set 2. And if the slice is bigger than the pizza... well, you have a magician's trick and Set 1 ⊇ Set 2!
In this case, without further information about the specific elements of the sets, we can't precisely determine the relationship between Set 1 and Set 2. But rest assured, the Venn diagram will help you visualize it in a more "circly" manner!
To determine the relationship between Set 1: (A - C) - (B - C) and Set 2: A - B using a Venn diagram, we first need to understand the basic concept of set operations.
The set operation symbols we will utilize are as follows:
- A - B represents the set difference, which means all the elements that are in set A but not in set B.
- A ⊆ B represents the subset relationship, meaning all the elements in set A are also in set B.
- A = B represents the equality relationship, indicating that set A and set B contain the exact same elements.
- A ⊇ B represents the superset relationship, implying that all the elements in set B are also in set A.
Let's proceed step-by-step with the Venn diagram:
Step 1: Draw two overlapping circles to represent the sets A and B.
________
/ A \
__/__________\__________
/ B \
Step 2: Within the circle representing set A, draw another smaller circle inside it to represent the set C.
________
/ A \
__/___ C __\__________
/ B \
Step 3: In the overlapping region between set A and set C, label it as (A - C).
________
/ A \
__/___ C __\__________
/ (A-C) B \
Step 4: In the region inside set B but outside set C, label it as (B - C).
________
/ A \
__/___ C __\__________
/ (A-C) (B-C)
Step 5: Finally, consider Set 1: (A - C) - (B - C) and Set 2: A - B.
- Set 1: (A - C) - (B - C) represents the elements that are in (A - C) but not in (B - C). This is the shaded region labeled (A-C).
- Set 2: A - B represents the elements that are in set A but not in set B. This region is labeled (A-C).
Since both sets have the same representation, we can conclude that Set 1: (A - C) - (B - C) = Set 2: A - B.
Thus, the relationship that is true for the pair of sets is = (equality relationship).
To determine the relationship between the sets Set 1: (A − C) − (B − C) and Set 2: A − B using a Venn diagram, we need to first understand the operations involved.
In set theory, the symbol "-" represents the set difference operation. This operation removes the elements in the second set from the first set.
Let's break down both sets step by step:
Set 1: (A − C) − (B − C)
1. (A − C): This means we take all the elements in set A that are not in set C.
2. (B − C): This means we take all the elements in set B that are not in set C.
3. (A − C) − (B − C): This means we take the result from step 1 and remove the elements from set B that are not in set C.
Set 2: A − B
Here, we take all the elements in set A that are not in set B.
To create a Venn diagram representing these sets, we need three circles: one for A, one for B, and one for C. The circles for A and B should overlap, as some elements are common to both sets.
Here's how we can proceed:
1. Draw a rectangle to represent the universal set that contains all the elements relevant to our problem.
2. Inside the rectangle, draw three circles representing A, B, and C. Make sure the circle for C is completely inside the rectangle, while the circles for A and B overlap.
Now, let's fill in the diagram based on the operations involved:
For Set 1: (A − C) − (B − C)
1. Start by shading the region inside the circle for C, as this represents the elements in C.
2. Next, shade the region in the circle for A but not in C, as this represents (A − C).
3. Now, shade the region in the circle for B but not in C, as this represents (B − C).
4. Finally, shade the region in the circle for A but not in B, which represents (A − C) − (B − C).
For Set 2: A − B
1. Shade the region in the circle for A but not in B, as this represents the elements in (A − B).
Now, compare the shaded regions in both diagrams.
If the shaded region for Set 1: (A − C) − (B − C) is entirely encompassed by the shaded region for Set 2: A − B, then we can conclude that ⊆ (subset relationship) holds. This is because every element in Set 1 is also in Set 2.
If the shaded regions for Set 1: (A − C) − (B − C) and Set 2: A − B are completely separate, without any overlap, then we can conclude that = (equality relationship) holds. This means the two sets have the exact same elements.
If the shaded region for Set 1: (A − C) − (B − C) encompasses the shaded region for Set 2: A − B, then we can conclude that ⊇ (superset relationship) holds. This is because Set 1 contains all the elements in Set 2, plus some additional elements.
By following these steps and visually comparing the shaded regions in the Venn diagram, you can determine which relationship (⊆, =, ⊇) is true for the pair of sets.