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Associative Law of Binary Operation

If A=1,2,3, B...

Question

If A = {1, 2, 3}, B = {4}, C = {5}, then verify that:

(i) A × (B ∪ C) = (A × B) ∪ (A × C)
(ii) A × (B ∩ C) = (A × B) ∩ (A × C)
(iii) A × (B − C) = (A × B) − (A × C)

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Solution

Given:
A = {1, 2, 3}, B = {4} and C = {5}

(i) A × (B ∪ C) = (A × B) ∪ (A × C)
We have:
(B ∪ C) = {4, 5}
LHS: A × (B ∪ C) = {(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)}
Now,
(A × B) = {(1, 4), (2, 4), (3, 4)}
And,
(A × C) = {(1, 5), (2, 5), (3, 5)}
RHS: (A × B) ∪ (A × C) = {(1, 4), (2, 4), (3, 4), (1, 5), (2, 5), (3, 5)}
∴ LHS = RHS

(ii) A × (B ∩ C) = (A × B) ∩ (A × C)
We have:
(B ∩ C) = $ϕ$
LHS: A × (B ∩ C) = $ϕ$
And,
(A × B) = {(1, 4), (2, 4), (3, 4)}
(A × C) = {(1, 5), (2, 5), (3, 5)}
RHS: (A × B) ∩ (A × C) = $ϕ$
∴ LHS = RHS

(iii) A × (B − C) = (A × B) − (A × C)
We have:
(B − C) = $ϕ$
LHS: A × (B − C) = $ϕ$
Now,
(A × B) = {(1, 4), (2, 4), (3, 4)}
And,
(A × C) = {(1, 5), (2, 5), (3, 5)}
RHS: (A × B) − (A × C) = $ϕ$
∴ LHS = RHS

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Similar questions

A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6}

D = {5, 6, 7, 8} .

(i) A \times (B \cap C) = (A \times B) \cap (A \times C)

(i i) A \times C

B \times D

If A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}, find

(i) $A \times (B \cap C)$

(ii) $(A \times B) \cap (A \times C)$

(iii) $A \times (B \cup C)$

(iv) $(A \times B) \cup (A \times C)$

Let A = {1,2,4,5}, B = {2,3,5,6} C = {4,5,6,7}. Verify the following identities :

$(i) A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$

(ii) $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$

(iii) $A \cap (B - C) = (A \cap B) - (A \cap C)$

(iv) $A - (B \cup C) = (A - B) \cap (A - C)$

(v) $A - (B \cap C) = (A - B) \cup (A - C)$

(vi) $A \cap (B Δ C) = (A \cap B) Δ (A \cap C)$

A \cup (B \cap C) = (A \cup B) \cap (A \cup C)

A \cap (B \cup C) = (A \cap B) \cup (A \cap C)

A \cap (B - C) = (A \cap B) - (A \cap C)

A - (B \cup C) = A (A - B) \cap (A - C)

A - (B \cap C) = (A - B) \cup (A - C)

A \cap (B ∆ C) = (A \cap B) ∆ (A \cap C)

Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that

(i) A × (B ∩ C) = (A × B) ∩ (A × C)

(ii) A × C is a subset of B × D

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