Prove that X ⊆ X U Y for all sets X and Y

• avdnowhere
In summary, The person is new to the forum and hopes to learn math more easily. They have a homework assignment due on Thursday and are looking for help on two questions. They mention having lecture notes, but they are different from the questions, so they are unsure how to start. The questions ask to prove a set relationship and a sequence identity. The expert suggests showing elements in the set and simplifying the sequence using algebra.
avdnowhere
hi guys...

this is my first thread in this forum..

i hope i'll learn math more easily with this forum...

mmm..

i've a some math homework that i must submit it this Thursday...

but, i don't know how to answer it...

the questions is...

1. Prove that X ⊆ X U Y for all sets X and Y...

2. A sequence r is defined as rn = 3.2ⁿ - 4.5ⁿ, n≥0.
Prove that the sequence satisfied rn = 7rn-1 - 10rn-2, n≥2.

anyone have the solution for these question?

You need to show what you have tried on homework problems. Then people will give you hints or suggestions when they see where you are having difficulty. Welcome to the forums.

LCKurtz said:
You need to show what you have tried on homework problems. Then people will give you hints or suggestions when they see where you are having difficulty. Welcome to the forums.

but the note is different from the question...

so i don't know how to start it...

avdnowhere said:
hi guys...

this is my first thread in this forum..

i hope i'll learn math more easily with this forum...

mmm..

i've a some math homework that i must submit it this Thursday...

but, i don't know how to answer it...

the questions is...

1. Prove that X ⊆ X U Y for all sets X and Y...

2. A sequence r is defined as rn = 3.2ⁿ - 4.5ⁿ, n≥0.
Prove that the sequence satisfied rn = 7rn-1 - 10rn-2, n≥2.

anyone have the solution for these question?

avdnowhere said:

but the note is different from the question...

so i don't know how to start it...

I will give you a couple hints. For the first one you must show every element in X is an element of X U Y. So start with an x in X and explain why it is in X U Y.

For the second one just calculate the right hand side and collect terms on powers of two and 5. It's straightforward algebra.

1. How do you prove that X is a subset of X U Y?

The most common way to prove this is by using the definition of a subset, which states that for all elements x in set X, x must also be in set X U Y. This can be shown through a logical argument or by using set notation.

2. Can you give an example to illustrate this proof?

Yes, for example, let X = {1, 2, 3} and Y = {3, 4, 5}. X U Y = {1, 2, 3, 4, 5}. Since all elements in X (1, 2, and 3) are also in X U Y, we can say that X is a subset of X U Y.

3. Is it possible for X to not be a subset of X U Y?

No, it is not possible. By definition, a set is always a subset of itself and since X U Y contains all elements of X, X must be a subset of X U Y.

4. Can this statement be proven using mathematical induction?

Yes, this statement can be proven using mathematical induction. The base case would be when X is an empty set, which is always a subset of any set. Then, for the inductive step, assume that X is a subset of X U Y for some set Y. Then, adding another element to X in the form of {x} does not change the fact that X is a subset of X U Y, since {x} is also in X U Y. Therefore, by mathematical induction, X is a subset of X U Y for all sets X and Y.

5. Are there any exceptions to this statement?

No, there are no exceptions to this statement. It holds true for all sets X and Y, regardless of their elements. This can be proven using the definition of a subset and logical arguments.

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