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Example 16: Let [tex]\( A = \{1, 2, 4\} \)[/tex], [tex]\( B = \{2, 4, 6, 8\} \)[/tex], and [tex]\( C = \{5, 7, 9\} \)[/tex]. Find:

a. [tex]\( A - B = \{1\} \)[/tex]

b. [tex]\( B - A = \{6, 8\} \)[/tex]

c. [tex]\( B - C = \{6, 8\} \)[/tex]

d. [tex]\( B \cup (C - A) = \)[/tex]

e. [tex]\( (A \cap C) - (B \cup A) = \)[/tex]

f. [tex]\( (B \cap C) - (C \cup A) = \)[/tex]

Sagot :

GinnyAnswer
Sure! Let's solve these set operations step-by-step, based on the given answer.

### Given Sets:
- [tex]\( A = \{1, 2, 4\} \)[/tex]
- [tex]\( B = \{2, 4, 6, 8\} \)[/tex]
- [tex]\( C = \{5, 7, 9\} \)[/tex]

### a. [tex]\( A - B \)[/tex]
This represents the set of elements that are in [tex]\( A \)[/tex] but not in [tex]\( B \)[/tex].
[tex]\[ A - B = \{1, 2, 4\} - \{2, 4, 6, 8\} = \{1\} \][/tex]

### b. [tex]\( B - A \)[/tex]
This represents the set of elements that are in [tex]\( B \)[/tex] but not in [tex]\( A \)[/tex].
[tex]\[ B - A = \{2, 4, 6, 8\} - \{1, 2, 4\} = \{6, 8\} \][/tex]

### c. [tex]\( B - C \)[/tex]
This represents the set of elements that are in [tex]\( B \)[/tex] but not in [tex]\( C \)[/tex].
[tex]\[ B - C = \{2, 4, 6, 8\} - \{5, 7, 9\} = \{2, 4, 6, 8\} \][/tex]

### d. [tex]\( B \cup (C - A) \)[/tex]
First, let's find [tex]\( C - A \)[/tex]:
[tex]\[ C - A = \{5, 7, 9\} - \{1, 2, 4\} = \{5, 7, 9\} \][/tex]

Now we calculate the union of [tex]\( B \)[/tex] and [tex]\( C - A \)[/tex]:
[tex]\[ B \cup \{5, 7, 9\} = \{2, 4, 6, 8\} \cup \{5, 7, 9\} = \{2, 4, 5, 6, 7, 8, 9\} \][/tex]

### e. [tex]\( (A \cap C) - (B \cup A) \)[/tex]
First, let's find [tex]\( A \cap C \)[/tex]:
[tex]\[ A \cap C = \{1, 2, 4\} \cap \{5, 7, 9\} = \emptyset \][/tex]

Now, we calculate [tex]\( B \cup A \)[/tex]:
[tex]\[ B \cup A = \{2, 4, 6, 8\} \cup \{1, 2, 4\} = \{1, 2, 4, 6, 8\} \][/tex]

Since [tex]\( A \cap C = \emptyset \)[/tex], any set minus the empty set will give the empty set.
[tex]\[ \emptyset - \{1, 2, 4, 6, 8\} = \emptyset \][/tex]

### f. [tex]\( (B \cap C) - (C \cup A) \)[/tex]
First, let's find [tex]\( B \cap C \)[/tex]:
[tex]\[ B \cap C = \{2, 4, 6, 8\} \cap \{5, 7, 9\} = \emptyset \][/tex]

Next, we calculate [tex]\( C \cup A \)[/tex]:
[tex]\[ C \cup A = \{5, 7, 9\} \cup \{1, 2, 4\} = \{1, 2, 4, 5, 7, 9\} \][/tex]

Since [tex]\( B \cap C = \emptyset \)[/tex], any set minus the empty set will give the empty set.
[tex]\[ \emptyset - \{1, 2, 4, 5, 7, 9\} = \emptyset \][/tex]

### Summary:
- a. [tex]\( A - B = \{1\} \)[/tex]
- b. [tex]\( B - A = \{6, 8\} \)[/tex]
- c. [tex]\( B - C = \{2, 4, 6, 8\} \)[/tex]
- d. [tex]\( B \cup (C - A) = \{2, 4, 5, 6, 7, 8, 9\} \)[/tex]
- e. [tex]\( (A \cap C) - (B \cup A) = \emptyset \)[/tex]
- f. [tex]\( (B \cap C) - (C \cup A) = \emptyset \)[/tex]
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