If $A$ and $B$ are any two sets, then state the following statement is true/false
$A - (A - B) = A \cap B$

True

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False

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Solution

The correct option is A True
$L . H . S = A - (A - B)$
$= A \cap (A - B)^{'}$ [ Set difference ]
$= A \cap (A \cap B^{'})^{'}$ [ Set difference ]
$= A \cap (A \cup (B^{'})^{'})$ [ De-Morgan's Law ]
$= A \cap (A^{'} \cup B)$ [ Double complement ]
$= (A \cap A^{'}) \cup (A \cap B)$ [ Distribution law ]
$= ϕ \cup (A \cap B)$ [ Since, A set and its complement are disjoint ]
$= A \cap B$
$= R . H . S$
$∴$ Given statement is true.

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Similar questions

If A and B are any two sets, then A ∪ (A ∩ B) is equal to A. State True/False.

State whether each of the following statement are true or false. If the statement is false, rewrite the given statement correctly.

(i) If P = {m, n} and Q = {n, m}, then P × Q = {(m, n), (n, m)}.

(ii) If A and B are non-empty sets, then A × B is a non-empty set of ordered pairs (x, y) such that x ∈ A and y ∈ B.

(iii) If A = {1, 2}, B = {3, 4}, then A × (B ∩ Φ) = Φ.

P = {m, n}

Q = {n, m}

P \times Q = {(m, n), (n, m)}

A

B

A \times B

(x, y)

x \in A

y \in B

A = {1, 2}, B = {3, 4}

A \times (B \cap ϕ) = ϕ

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