Chapter 3: Sets and Functions

Solve · page 56: 4. Verify the commutative properties of union and intersection for the following pairs of sets: (1) A= {1,2,3,4,5}, B= {4, 6, 8, 10} Gi) NZ Gi) A={x|x e RA x>0}, BER.

(i)

Solution $$\text{LHS} = A \cup B$$ $$\text{LHS} = \{1, 2, 3, 4, 5\} \cup \{4, 6, 8, 10\}$$ $$\text{LHS} = \{1, 2, 3, 4, 5, 6, 8, 10\}$$ $$\text{RHS} = B \cup A$$ $$\text{RHS} = \{4, 6, 8, 10\} \cup \{1, 2, 3, 4, 5\}$$ $$\text{RHS} = \{1, 2, 3, 4, 5, 6, 8, 10\}$$ $$\text{LHS} = \text{RHS}$$ $$\text{LHS} = A \cap B$$ $$\text{LHS} = \{1, 2, 3, 4, 5\} \cap \{4, 6, 8, 10\}$$ $$\text{LHS} = \{4\}$$ $$\text{RHS} = B \cap A$$ $$\text{RHS} = \{4, 6, 8, 10\} \cap \{1, 2, 3, 4, 5\}$$ $$\text{RHS} = \{4\}$$ $$\text{LHS} = \text{RHS}$$ Hence, it is verified that the commutative properties of union [مباح تبادلی خاصیتِ اتحاد] and intersection [مباح تبادلی خاصیتِ تقاطع] hold for the given sets.

Explanation

• Write the left-hand side expression for union.
• Substitute the elements of sets $A$ and $B$ into the expression.
• Combine all elements from both sets without repetition to find the union.
• Write the right-hand side expression for union.
• Substitute the elements of sets $B$ and $A$ into the expression.
• Combine all elements from both sets without repetition to find the union.
• Compare both sides to show they are equal.
• Write the left-hand side expression for intersection.
• Substitute the elements of sets $A$ and $B$ into the expression.
• Find the common elements between both sets to find the intersection.
• Write the right-hand side expression for intersection.
• Substitute the elements of sets $B$ and $A$ into the expression.
• Find the common elements between both sets to find the intersection.
• Compare both sides to show they are equal.

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(ii)

Solution $$\text{LHS} = N \cup Z$$ $$\text{LHS} = \{1, 2, 3, 4, \dots\} \cup \{0, \pm 1, \pm 2, \pm 3, \dots\}$$ $$\text{LHS} = \{0, \pm 1, \pm 2, \pm 3, \dots\}$$ $$\text{RHS} = Z \cup N$$ $$\text{RHS} = \{0, \pm 1, \pm 2, \pm 3, \dots\} \cup \{1, 2, 3, 4, \dots\}$$ $$\text{RHS} = \{0, \pm 1, \pm 2, \pm 3, \dots\}$$ $$\text{LHS} = \text{RHS}$$ $$\text{LHS} = N \cap Z$$ $$\text{LHS} = \{1, 2, 3, 4, \dots\} \cap \{0, \pm 1, \pm 2, \pm 3, \dots\}$$ $$\text{LHS} = \{1, 2, 3, 4, \dots\}$$ $$\text{RHS} = Z \cap N$$ $$\text{RHS} = \{0, \pm 1, \pm 2, \pm 3, \dots\} \cap \{1, 2, 3, 4, \dots\}$$ $$\text{RHS} = \{1, 2, 3, 4, \dots\}$$ $$\text{LHS} = \text{RHS}$$ Hence, it is verified that the commutative properties of union [مباح تبادلی خاصیتِ اتحاد] and intersection [مباح تبادلی خاصیتِ تقاطع] hold for the given sets.

Explanation

• Write the left-hand side expression for union.
• Substitute the descriptive or roster forms of sets of natural numbers $N$ and integers $Z$.
• Combine both sets to get the larger set $Z$.
• Write the right-hand side expression for union.
• Substitute the roster forms of sets $Z$ and $N$.
• Combine both sets to get the set $Z$.
• Compare both sides to show they are equal.
• Write the left-hand side expression for intersection.
• Substitute the roster forms of sets $N$ and $Z$.
• Find the common elements between both sets, which form the set $N$.
• Write the right-hand side expression for intersection.
• Substitute the roster forms of sets $Z$ and $N$.
• Find the common elements between both sets, which form the set $N$.
• Compare both sides to show they are equal.

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(iii)

Solution $$\text{LHS} = A \cup B$$ $$\text{LHS} = \{x \mid x \in R \land x \ge 0\} \cup R$$ $$\text{LHS} = R$$ $$\text{RHS} = B \cup A$$ $$\text{RHS} = R \cup \{x \mid x \in R \land x \ge 0\}$$ $$\text{RHS} = R$$ $$\text{LHS} = \text{RHS}$$ $$\text{LHS} = A \cap B$$ $$\text{LHS} = \{x \mid x \in R \land x \ge 0\} \cap R$$ $$\text{LHS} = \{x \mid x \in R \land x \ge 0\}$$ $$\text{RHS} = B \cap A$$ $$\text{RHS} = R \cap \{x \mid x \in R \land x \ge 0\}$$ $$\text{RHS} = \{x \mid x \in R \land x \ge 0\}$$ $$\text{LHS} = \text{RHS}$$ Hence, it is verified that the commutative properties of union [مباح تبادلی خاصیتِ اتحاد] and intersection [مباح تبادلی خاصیتِ تقاطع] hold for the given sets.

Explanation

• Write the left-hand side expression for union.
• Substitute the set-builder form of set $A$ and set $R$.
• Combine the subset of non-negative real numbers with the set of all real numbers, yielding $R$.
• Write the right-hand side expression for union.
• Substitute the set $R$ and set-builder form of set $A$.
• Combine the sets to get $R$.
• Compare both sides to show they are equal.
• Write the left-hand side expression for intersection.
• Substitute the set-builder form of set $A$ and set $R$.
• Find the common elements between the set of all real numbers and its non-negative subset, yielding set $A$.
• Write the right-hand side expression for intersection.
• Substitute set $R$ and the set-builder form of set $A$.
• Find the common elements between the sets, yielding set $A$.
• Compare both sides to show they are equal.