iiamiva
Answered

At Westonci.ca, we make it easy for you to get the answers you need from a community of knowledgeable individuals. Experience the convenience of getting accurate answers to your questions from a dedicated community of professionals. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.

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.

Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.