Question

Prove that if A and B are any two sets such that A ⊆ B, then A ∪ B = B
i) by direct method;
ii) by proving its contrapositive;
iii) by contradiction.

Answer 1

Answer on Question #44546 – Math - Algebra

Problem.

Prove that if $A$ and $B$ are any two sets such that $A \subseteq B$, then $A \cup B = B$.

i) by direct method;

ii) by proving its contrapositive;

iii) by contradiction.

Solution.

i) If $x \in A \cup B$, then $x \in A$ or $x \in B$. Hence $x \in B$, as $A \subseteq B$. Therefore $A \cup B \subseteq B$.

If $x \in B$, then $x \in A \cup B$ (because $A \subseteq B$). Therefore $B \subseteq A \cup B$.

Since $A \cup B = B$, as $A \cup B \subseteq B$ and $B \subseteq A \cup B$.

ii) We need to prove that if $A \cup B \neq B$, then $A \not\subseteq B$.

If $A \cup B \neq B$, then there exists $x \in A$ such that $x \notin B$. Therefore $A \not\subseteq B$.

iii) Suppose that $A \subseteq B$ and $A \cup B \neq B$. If $A \cup B \neq B$, then there exists $x \in A$ such that $x \notin B$. Therefore $A \not\subseteq B$. We obtain a contradiction with assumption $A \subseteq B$.

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