Question

Let ,, and be sets. Prove that if ⊆ then − ⊆−.

Answer 1

If
`\(A \subseteq B\)`, then
`\(B - C \subseteq A - C\)`.

Step-by-step explanation:

The statement
`\(A \subseteq B\)` means that every element in set (A) is also an element of set (B). Now, let's consider the set difference (B - C), which consists of elements that are in (B) but not in (C). Since every element in (A) is in (B) (according to
`\(A \subseteq B\)`, it follows that every element in (B - C) is also in (A). Therefore,
`\(B - C \subseteq A\)`. Similarly, (A - C) is the set of elements in (A) but not in (C). Since
`\(B - C \subseteq A\)`, it implies that (B - C) is a subset of the elements in (A - C), concluding the proof.

In mathematical terms, let (x) be an arbitrary element in (B - C). This means
`\(x \in B\)` and
`\(x \\otin C\)`. Since
`\(A \subseteq B\)` ,
`\(x \in A\)`. Now, considering (x) in the context of (A - C), it satisfies
`\(x \in A\)` and
`\(x \\otin C\)`. Therefore,
`\(x \in A - C\)`, proving that
`\(B - C \subseteq A - C\).` This logical deduction establishes the validity of the original statement, completing the proof.

Question:

Let (A), (B), and (C) be sets. Prove that if
`\(A \subseteq B\)`, then
`\(B - C \subseteq A - C\).`

Answer 2

If
`\(A \subseteq B\)`, then
`\(A^c \subseteq B^c\)`.

Step-by-step explanation:

The given statement can be proved using set theory principles. Let's denote
`\(A\)` and
`\(B\)` as sets. The statement
`\(A \subseteq B\)` means that every element in set
`\(A\)` is also an element of set
`\(B\)`. Now, we need to prove that if
`\(A \subseteq B\)`, then
`\(A^c \subseteq B^c\)`, where
`\(A^c\)` and
`\(B^c\)` are the complements of sets
`\(A\)` and
`\(B\)`, respectively.

The complement of a set contains all elements not in the original set. So, if
`\(A \subseteq B\)`, then every element not in
`\(A\)` is also not in
`\(B\)`. This implies that every element in
`\(A^c\)` is in
`\(B^c\)`, proving that
`\(A^c \subseteq B^c\)`.

In mathematical terms:

`\[A \subseteq B \implies x \in A \implies x \\otin A^c \implies x \\otin B^c \implies x \in B^c \implies A^c \subseteq B^c.\]`

Therefore, the statement
`\(A \subseteq B \implies A^c \subseteq B^c\)` is true, confirming the given proposition.