Distributive property — lesson. Mathematics State Board, Class 9.

UPSKILL MATH PLUS

Learn Mathematics through our AI based learning portal with the support of our Academic Experts!

Learn more

Distributive property of intersection over union

For any three sets

$A$ ,

$B$ and

$C$ :

$A \cap (B \cup C)$

$=$

$(A \cap B)$

$\cup$

$(A \cap C)$

Let

$A$

$=$

$\{$

$l$ ,

$m$ ,

$n$ ,

$o$ ,

$p$

$\}$ ,

$B$

$=$

$\{$

$n$ ,

$o$ ,

$p$ ,

$q$ ,

$r$

$\}$ and

$C$

$=$

$\{$

$l$ ,

$n$ ,

$p$ ,

$r$

$\}$

L.H.S:

$A \cap (B \cup C)$

$B \cup C$

$=$

$\{$

$n$ ,

$o$ ,

$p$ ,

$q$ ,

$r$

$\}$

$\cup$

$\{$

$l$ ,

$n$ ,

$p$ ,

$r$

$\}$

$B \cup C$

$=$

$\{$

$l$ ,

$n$ ,

$o$ ,

$p$ ,

$q$ ,

$r$

$\}$

$A \cap (B \cup C)$

$=$

$\{$

$l$ ,

$m$ ,

$n$ ,

$o$ ,

$p$

$\}$

$\cap$

$\{$

$l$ ,

$n$ ,

$o$ ,

$p$ ,

$q$ ,

$r$

$\}$

$A \cap (B \cup C)$

$=$

$\{$

$l$ ,

$n$ ,

$o$ ,

$p$

$\}$ - - - - - - - - - (I)

R.H.S:

$(A \cap B)$

$\cup$

$(A \cap C)$

$A \cap B$

$=$

$\{$

$l$ ,

$m$ ,

$n$ ,

$o$ ,

$p$

$\}$

$\cap$

$\{$

$n$ ,

$o$ ,

$p$ ,

$q$ ,

$r$

$\}$

$A \cap B$

$=$

$\{$

$n$ ,

$o$ ,

$p$

$\}$

$A \cap C$

$=$

$\{$

$l$ ,

$m$ ,

$n$ ,

$o$ ,

$p$

$\}$

$\cap$

$\{$

$l$ ,

$n$ ,

$p$ ,

$r$

$\}$

$A \cap C$

$=$

$\{$

$l$ ,

$n$ ,

$p$

$\}$

$(A \cap B)$

$\cup$

$(A \cap C)$

$=$

$\{$

$n$ ,

$o$ ,

$p$

$\}$

$\cup$

$\{$

$l$ ,

$n$ ,

$p$

$\}$

$(A \cap B)$

$\cup$

$(A \cap C)$

$=$

$\{$

$l$ ,

$n$ ,

$o$ ,

$p$

$\}$ - - - - - - - - - (II)

From (I) and (II), we see that:

$A \cap (B \cup C)$

$=$

$(A \cap B)$

$\cup$

$(A \cap C)$

This is called distributive property of intersection over union.

Distributive property of union over intersection

For any three sets

$A$ ,

$B$ and

$C$ :

$A \cup (B \cap C)$

$=$

$(A \cup B)$

$\cap$

$(A \cup C)$

Let

$A$

$=$

$\{$

$l$ ,

$m$ ,

$n$ ,

$o$ ,

$p$

$\}$ ,

$B$

$=$

$\{$

$n$ ,

$o$ ,

$p$ ,

$q$ ,

$r$

$\}$ and

$C$

$=$

$\{$

$l$ ,

$n$ ,

$p$ ,

$r$

$\}$

L.H.S:

$A \cup (B \cap C)$

$B \cap C$

$=$

$\{$

$n$ ,

$o$ ,

$p$ ,

$q$ ,

$r$

$\}$

$\cap$

$\{$

$l$ ,

$n$ ,

$p$ ,

$r$

$\}$

$B \cap C$

$=$

$\{$

$n$ ,

$p$ ,

$r$

$\}$

$A \cup (B \cap C)$

$=$

$\{$

$l$ ,

$m$ ,

$n$ ,

$o$ ,

$p$

$\}$

$\cup$

$\{$

$n$ ,

$p$ ,

$r$

$\}$

$A \cup (B \cap C)$

$=$

$\{$

$l$ ,

$m$ ,

$n$ ,

$o$ ,

$p$ ,

$r$

$\}$ - - - - - - - - - (I)

R.H.S:

$(A \cup B)$

$\cap$

$(A \cup C)$

$A \cup B$

$=$

$\{$

$l$ ,

$m$ ,

$n$ ,

$o$ ,

$p$

$\}$

$\cup$

$\{$

$n$ ,

$o$ ,

$p$ ,

$q$ ,

$r$

$\}$

$A \cup B$

$=$

$\{$

$l$ ,

$m$ ,

$n$ ,

$o$ ,

$p$ ,

$q$ ,

$r$

$\}$

$A \cup C$

$=$

$\{$

$l$ ,

$m$ ,

$n$ ,

$o$ ,

$p$

$\}$

$\cup$

$\{$

$l$ ,

$n$ ,

$p$ ,

$r$

$\}$

$A \cup C$

$=$

$\{$

$l$ ,

$m$ ,

$n$ ,

$o$ ,

$p$ ,

$r$

$\}$

$(A \cup B)$

$\cap$

$(A \cup C)$

$=$

$\{$

$l$ ,

$m$ ,

$n$ ,

$o$ ,

$p$ ,

$q$ ,

$r$

$\}$

$\cap$

$\{$

$l$ ,

$m$ ,

$n$ ,

$o$ ,

$p$ ,

$r$

$\}$

$(A \cup B)$

$\cap$

$(A \cup C)$

$=$

$\{$

$l$ ,

$m$ ,

$n$ ,

$o$ ,

$p$ ,

$r$

$\}$ - - - - - - - - - (II)

From (I) and (II), we see that:

$A \cup (B \cap C)$

$=$

$(A \cup B)$

$\cap$

$(A \cup C)$

This is called distributive property of intersection over union.

Important!

L.H.S – Left Hand Side

R.H.S – Right Hand Side

Previous theory

Exit to the topic

Next theory

Send feedback

Did you find an error?

Send it to us!

3. Distributive property

Theory: