Name: Binary Operations
Uploaded: 2019-07-19
Duration: 9 min 24 s
Description: What are the binary operations? What are the terms associated with it? Where do we use binary operations? Watch out for this video to learn more about this c...

Question

For any sets $A, B$ and $C$ , prove that $A \times (B - C) = (A \times B) - (A \times C)$

Accepted Answer

Given $A, B$ and $C$ are three sets.

We need to prove: $A \times (B - C) = (A \times B) - (A \times C)$

Let us consider $(x, y) \in A \times (B - C)$

$\Rightarrow x \in A and y \in (B - C)$

$\Rightarrow x \in A and (y \in B and y \notin C)$

$\Rightarrow (x \in A and y \in B) and (x \in A and y \notin C)$

$\Rightarrow (x, y) \in (A \times B) and (x, y) \notin (A \times C)$

$\Rightarrow (x, y) \in (A \times B) - (A \times C)$ from this we can conclude that,

$\Rightarrow A \times (B - C) \subseteq (A \times B) - (A \times C) . . . (1)$

Let us consider again, $(a, b) \in (A \times B) - (A \times C)$

$\Rightarrow (a, b) \in (A \times B) and (a, b) \notin (A \times C)$

$\Rightarrow (a \in A and b \in B) and (a \in A and b \notin C)$

$\Rightarrow a \in A and (b \in B and b \notin C)$

$\Rightarrow a \in A and b \in (B - C)$

$\Rightarrow (a, b) \in A \times (B - C)$ from this, we can conclude that,

$\Rightarrow (A \times B) - (A \times C) \subseteq A \times (B - C) . . . (2)$

Now by the definition of the set, we can say that, from $(1)$ and $(2)$ ,

$A \times (B - C) = (A \times B) - (A \times C)$ [Proved]

For any sets $A, B$ and $C$ , prove that $A \times (B - C) = (A \times B) - (A \times C)$

Important Questions on Relations

Learn and Prepare for any exam you want !

For any sets A,B and C, prove that A×(B-C)=(A×B)-(A×C)

Important Questions on Relations

Learn and Prepare for any exam you want !

For any sets $A, B$ and $C$ , prove that $A \times (B - C) = (A \times B) - (A \times C)$