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EXERCISE 1.1
Let P and Q are the subsets of a universal set U. Write the set operations defined by
the following set-builder forms.
a) {x: x = P or xЄ Q}
d) {x: x = Q but xe P}
b) {x: xЄ P and x = Q}
e) {x: xЄ U but x = P}
g) {x: x = U but x P or Q} h) {x: xe U but x P and Q}
c) {x: xЄ P but x Q}
f) {x: xЄ U but x £Q}
i) {x: xЄ U but x P - Q}



Answer :

GinnyAnswer
Let's go through each of the set operations described by their set-builder notation step-by-step:

### Given:
- [tex]\( P \)[/tex] and [tex]\( Q \)[/tex] are subsets of a universal set [tex]\( U \)[/tex].

### a) [tex]\(\{x \mid x = P \text{ or } x \in Q \}\)[/tex]
This represents the union of sets [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]. It includes all elements that are in [tex]\( P \)[/tex] or [tex]\( Q \)[/tex] or both.

Result: [tex]\( P \cup Q \)[/tex]

### b) [tex]\(\{x \mid x \in P \text{ and } x = Q \}\)[/tex]
This is slightly confusing due to the notation. Assuming it intends to mean 'and' in the regular sense, it should be the intersection of [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]. This includes all elements that are in both [tex]\( P \)[/tex] and [tex]\( Q \)[/tex].

Result: [tex]\( P \cap Q \)[/tex]

### c) [tex]\(\{x \mid x \in P \text{ but } x \notin Q \}\)[/tex]
This represents the difference between sets [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]. It includes all elements that are in [tex]\( P \)[/tex] but not in [tex]\( Q \)[/tex].

Result: [tex]\( P - Q \)[/tex]

### d) [tex]\(\{x \mid x = Q \text{ but } x \notin P \}\)[/tex]
This represents the difference between sets [tex]\( Q \)[/tex] and [tex]\( P \)[/tex]. It includes all elements that are in [tex]\( Q \)[/tex] but not in [tex]\( P \)[/tex].

Result: [tex]\( Q - P \)[/tex]

### e) [tex]\(\{x \mid x \in U \text{ but } x = P \}\)[/tex]
This represents the complement of [tex]\( P \)[/tex] in [tex]\( U \)[/tex]. It includes all elements that are in [tex]\( U \)[/tex] but not in [tex]\( P \)[/tex].

Result: [tex]\( U - P \)[/tex]

### f) [tex]\(\{x \mid x \in U \text{ but } x \notin Q \}\)[/tex]
This represents the complement of [tex]\( Q \)[/tex] in [tex]\( U \)[/tex]. It includes all elements that are in [tex]\( U \)[/tex] but not in [tex]\( Q \)[/tex].

Result: [tex]\( U - Q \)[/tex]

### g) [tex]\(\{x \mid x = U \text{ but } x \notin (P \text{ or } Q) \}\)[/tex]
This represents the difference between the universal set [tex]\( U \)[/tex] and the union of [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]. It includes all elements that are in [tex]\( U \)[/tex] but not in [tex]\( P \)[/tex] or [tex]\( Q \)[/tex].

Result: [tex]\( U - (P \cup Q) \)[/tex]

### h) [tex]\(\{x \mid x \in U \text{ but } x \notin (P \cap Q) \}\)[/tex]
This represents the difference between the universal set [tex]\( U \)[/tex] and the intersection of [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]. It includes all elements that are in [tex]\( U \)[/tex] but not in both [tex]\( P \)[/tex] and [tex]\( Q \)[/tex].

Result: [tex]\( U - (P \cap Q) \)[/tex]

### i) [tex]\(\{x \mid x \in U \text{ but } x \notin (P - Q) \}\)[/tex]
This represents the difference between the universal set [tex]\( U \)[/tex] and [tex]\( P \)[/tex] minus [tex]\( Q \)[/tex]. It includes all elements that are in [tex]\( U \)[/tex] but not in the difference between [tex]\( P \)[/tex] and [tex]\( Q \)[/tex].

Result: [tex]\( U - (P - Q) \)[/tex]

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### Summary of Results
a) [tex]\( P \cup Q \)[/tex]

b) [tex]\( P \cap Q \)[/tex]

c) [tex]\( P - Q \)[/tex]

d) [tex]\( Q - P \)[/tex]

e) [tex]\( U - P \)[/tex]

f) [tex]\( U - Q \)[/tex]

g) [tex]\( U - (P \cup Q) \)[/tex]

h) [tex]\( U - (P \cap Q) \)[/tex]

i) [tex]\( U - (P - Q) \)[/tex]

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