iiamiva
Answered

Westonci.ca is your trusted source for accurate answers to all your questions. Join our community and start learning today! Explore a wealth of knowledge from professionals across different disciplines on our comprehensive platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.

prove A-(BnC) = (A-B)U(A-C), explain with an example​

Sagot :

jacob193

Answer:

Prove set equality by showing that for any element [tex]x[/tex], [tex]x \in (A \backslash (B \cap C))[/tex] if and only if [tex]x \in ((A \backslash B) \cup (A \backslash C))[/tex].

Example:

[tex]A = \lbrace 0,\, 1,\, 2,\, 3 \rbrace[/tex].

[tex]B = \lbrace0,\, 1 \rbrace[/tex].

[tex]C = \lbrace0,\, 2 \rbrace[/tex].

[tex]\begin{aligned} & A \backslash (B \cap C) \\ =\; & \lbrace 0,\, 1,\, 2,\, 3 \rbrace \backslash \lbrace 0 \rbrace \\ =\; & \lbrace 1,\, 2,\, 3 \rbrace \end{aligned}[/tex].

[tex]\begin{aligned}& (A \backslash B) \cup (A \backslash C) \\ =\; & \lbrace 2,\, 3\rbrace \cup \lbrace 1,\, 3 \rbrace \\ =\; & \lbrace 1,\, 2,\, 3 \rbrace\end{aligned}[/tex].

Step-by-step explanation:

Proof for [tex][x \in (A \backslash (B \cap C))] \implies [x \in ((A \backslash B) \cup (A \backslash C))][/tex] for any element [tex]x[/tex]:

Assume that [tex]x \in (A \backslash (B \cap C))[/tex]. Thus, [tex]x \in A[/tex] and [tex]x \not \in (B \cap C)[/tex].

Since [tex]x \not \in (B \cap C)[/tex], either [tex]x \not \in B[/tex] or [tex]x \not \in C[/tex] (or both.)

  • If [tex]x \not \in B[/tex], then combined with [tex]x \in A[/tex], [tex]x \in (A \backslash B)[/tex].
  • Similarly, if [tex]x \not \in C[/tex], then combined with [tex]x \in A[/tex], [tex]x \in (A \backslash C)[/tex].

Thus, either [tex]x \in (A \backslash B)[/tex] or [tex]x \in (A \backslash C)[/tex] (or both.)

Therefore, [tex]x \in ((A \backslash B) \cup (A \backslash C))[/tex] as required.

Proof for [tex][x \in ((A \backslash B) \cup (A \backslash C))] \implies [x \in (A \backslash (B \cap C))][/tex]:

Assume that [tex]x \in ((A \backslash B) \cup (A \backslash C))[/tex]. Thus, either [tex]x \in (A \backslash B)[/tex] or [tex]x \in (A \backslash C)[/tex] (or both.)

  • If [tex]x \in (A \backslash B)[/tex], then [tex]x \in A[/tex] and [tex]x \not \in B[/tex]. Notice that [tex](x \not \in B) \implies (x \not \in (B \cap C))[/tex] since the contrapositive of that statement, [tex](x \in (B \cap C)) \implies (x \in B)[/tex], is true. Therefore, [tex]x \not \in (B \cap C)[/tex] and thus [tex]x \in A \backslash (B \cap C)[/tex].
  • Otherwise, if [tex]x \in A \backslash C[/tex], then [tex]x \in A[/tex] and [tex]x \not \in C[/tex]. Similarly, [tex]x \not \in C \![/tex] implies [tex]x \not \in (B \cap C)[/tex]. Therefore, [tex]x \in A \backslash (B \cap C)[/tex].

Either way, [tex]x \in A \backslash (B \cap C)[/tex].

Therefore, [tex]x \in ((A \backslash B) \cup (A \backslash C))[/tex] implies [tex]x \in A \backslash (B \cap C)[/tex], as required.

We appreciate your time. Please come back anytime for the latest information and answers to your questions. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.