To prove that f(E ∪ F) = f(E) ∪ f(F), we need to show that each set is a subset of the other.

Step 1: Show that f(E ∪ F) is a subset of f(E) ∪ f(F)

Let x be an element of f(E ∪ F). This means that there exists an element y in E ∪ F such that f(y) = x. Since y is in E ∪ F, y is in E or y is in F. If y is in E, then x is in f(E). If y is in F, then x is in f(F). Therefore, in either case, x is in f(E) ∪ f(F). This shows that every element of f(E ∪ F) is also an element of f(E) ∪ f(F), so f(E ∪ F) is a subset of f(E) ∪ f(F).

Step 2: Show that f(E) ∪ f(F) is a subset of f(E ∪ F)

Let x be an element of f(E) ∪ f(F). This means that x is in f(E) or x is in f(F). If x is in f(E), then there exists an element y in E such that f(y) = x. Since E is a subset of E ∪ F, y is also in E ∪ F, so x is in f(E ∪ F). Similarly, if x is in f(F), then there exists an element y in F such that f(y) = x. Since F is a subset of E ∪ F, y is also in E ∪ F, so x is in f(E ∪ F). Therefore, in either case, x is in f(E ∪ F). This shows that every element of f(E) ∪ f(F) is also an element of f(E ∪ F), so f(E) ∪ f(F) is a subset of f(E ∪ F).

Since each set is a subset of the other, we can conclude that f(E ∪ F) = f(E) ∪ f(F).

Question

To prove that f(E ∪ F) = f(E) ∪ f(F), we need to show that each set is a subset of the other.

Step 1: Show that f(E ∪ F) is a subset of f(E) ∪ f(F)

Let x be an element of f(E ∪ F). This means that there exists an element y in E ∪ F such that f(y) = x. Since y is in E ∪ F, y is in E or y is in F. If y is in E, then x is in f(E). If y is in F, then x is in f(F). Therefore, in either case, x is in f(E) ∪ f(F). This shows that every element of f(E ∪ F) is also an element of f(E) ∪ f(F), so f(E ∪ F) is a subset of f(E) ∪ f(F).

Step 2: Show that f(E) ∪ f(F) is a subset of f(E ∪ F)

Let x be an element of f(E) ∪ f(F). This means that x is in f(E) or x is in f(F). If x is in f(E), then there exists an element y in E such that f(y) = x. Since E is a subset of E ∪ F, y is also in E ∪ F, so x is in f(E ∪ F). Similarly, if x is in f(F), then there exists an element y in F such that f(y) = x. Since F is a subset of E ∪ F, y is also in E ∪ F, so x is in f(E ∪ F). Therefore, in either case, x is in f(E ∪ F). This shows that every element of f(E) ∪ f(F) is also an element of f(E ∪ F), so f(E) ∪ f(F) is a subset of f(E ∪ F).

Since each set is a subset of the other, we can conclude that f(E ∪ F) = f(E) ∪ f(F).

Knowee AI · Accepted Answer

To prove that f(E ∪ F) = f(E) ∪ f(F), we need to show that each set is a subset of the other.

Step 1: Show that f(E ∪ F) is a subset of f(E) ∪ f(F)

Let x be an element of f(E ∪ F). This means that there exists an element y in E ∪ F such that f(y) = x. Since y is in E ∪ F, y is in E or y is in F. If y is in E, then x is in f(E). If y is in F, then x is in f(F). Therefore, in either case, x is in f(E) ∪ f(F). This shows that every element of f(E ∪ F) is also an element of f(E) ∪ f(F), so f(E ∪ F) is a subset of f(E) ∪ f(F).

Step 2: Show that f(E) ∪ f(F) is a subset of f(E ∪ F)

Let x be an element of f(E) ∪ f(F). This means that x is in f(E) or x is in f(F). If x is in f(E), then there exists an element y in E such that f(y) = x. Since E is a subset of E ∪ F, y is also in E ∪ F, so x is in f(E ∪ F). Similarly, if x is in f(F), then there exists an element y in F such that f(y) = x. Since F is a subset of E ∪ F, y is also in E ∪ F, so x is in f(E ∪ F). Therefore, in either case, x is in f(E ∪ F). This shows that every element of f(E) ∪ f(F) is also an element of f(E ∪ F), so f(E) ∪ f(F) is a subset of f(E ∪ F).

Since each set is a subset of the other, we can conclude that f(E ∪ F) = f(E) ∪ f(F).

(ii) Prove thatf (E ∪ F ) = f (E) ∪ f (F )

Question

Solution

Similar Questions

Upgrade your grade with Knowee