Show that A - B-A - C need not imply B-C

Let A = 1, 2, 3, 4, B = 2, 3, 4, 5, 6, C = 2, 3, 4, 9, 10. ∴ A∩ B = 1, 2, 3, 4 ∩ 2, 3, 4, 5, 6 = 2, 3, 4 A∩ C = 1, 2, 3, 4 ∩ 2, 3, 4, 9, 10 = 2, 3, ...

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Standard XII

Mathematics

Complement of an Event

Show that A ∩...

Question

Show that $A \cap B = A \cap C$ need not imply $B = C .$

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Solution

Let A = {1, 2, 3, 4}, B = {2, 3, 4, 5, 6}, C = {2, 3, 4, 9, 10}.

$∴$ $A \cap B$ = {1, 2, 3, 4} $\cap$ {2, 3, 4, 5, 6}

= {2, 3, 4}

$A \cap C$ = {1, 2, 3, 4} $\cap$ {2, 3, 4, 9, 10}

= {2, 3, 4}

Now we have

$A \cap B = A \cap C$

But $B \neq C .$

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Show that A∩B=A∩C need not imply B=C.

Let A = {1, 2, 3, 4}, B = {2, 3, 4, 5, 6}, C = {2, 3, 4, 9, 10}. ∴ A∩B = {1, 2, 3, 4} ∩ {2, 3, 4, 5, 6} = {2, 3, 4} A∩C = {1, 2, 3, 4} ∩ {2, 3, 4, 9, 10} = {2, 3, 4} Now we have A∩B=A∩C But B≠C.

Show that $A \cap B = A \cap C$ need not imply $B = C .$

Let A = {1, 2, 3, 4}, B = {2, 3, 4, 5, 6}, C = {2, 3, 4, 9, 10}.

$∴$ $A \cap B$ = {1, 2, 3, 4} $\cap$ {2, 3, 4, 5, 6}

= {2, 3, 4}

$A \cap C$ = {1, 2, 3, 4} $\cap$ {2, 3, 4, 9, 10}

= {2, 3, 4}

Now we have

$A \cap B = A \cap C$

But $B \neq C .$