Disprove the following statement by ﬁnding a counterexample. $\text { For all sets } A , B \text {, and } C , A \cup ( B \cap C ) \subseteq ( A \cup B ) \cap C \text {. }$

Name: Disprove the following statement by ﬁnding a counterexample. $\text { For all sets } A , B \text {, and } C , A \cup ( B \cap C ) \subseteq ( A \cup B ) \cap C \text {. }$
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Question

Quizplus · Accepted Answer

The Answer is Counterexample: Let

A = \{ 1 \} , B = \{ 2 \}

, and

C = \{ 2 \}

. Then

A \cup ( B \cap C ) = \{ 1 \} \cup ( \{ 2 \} \cap \{ 2 \} ) =

\{ 1 \} \cup \{ 2 \} = \{ 1,2 \}

,
and

( A \cup B ) \cap C = ( \{ 1 \} \cup \{ 2 \} ) \cap \{ 2 \} = \{ 1,2 \} \cap \{ 2 \} = \{ 2 \}

. Thus

1 \in A \cup ( B \cap C )

whereas

1 otin ( A \cup B ) \cap C

. Therefore,

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

.
[This is one counterexample among many.]

Disprove the Following Statement by ﬁNding a Counterexample For all sets A,B, and C,A∪(B∩C)⊆(A∪B)∩C. \text { For all sets } A , B \text {, and } C , A \cup ( B \cap C ) \subseteq ( A \cup B ) \cap C \text {. } For all sets A,B, and C,A∪(B∩C)⊆(A∪B)∩C.

Disprove the Following Statement by ﬁNding a Counterexample $\text { For all sets } A , B \text {, and } C , A \cup ( B \cap C ) \subseteq ( A \cup B ) \cap C \text {. }$