2. Let A-(a ER Va E R(2-4z a)). Describe A. 3. Is the statement true or false? P

Question

2. Let A-(a ER Va E R(2-4z a)). Describe A. 3. Is the statement true or false? Prove your answer. (b) Let f : Z Z defined by f(x)-3r3-r. Then (i) f in injective (one-to-one). (ii) f is surjective (onto). 4. Decide whether the statement is true or false. Prove your answer. (a) For any two sets A and B, (AnBy Ac UBe (b) For any three sets A, B, C, if A BUC, then A B-C. 5. For a, b e R, define a ~ b if and only if a -beZ. (a) Prove that ~is an equivalence relation on Z. (b) Describe the equivalence class containing 4. (c) List 3 miembers of the equivalence class containing 6. Let n > 3 and A = { 1, 2, 3, 4, . . . ,n). How many subsets B of A contain both i and 3? Prove your answer.

Explanation / Answer

3.b. to prove the function is injective and surjective,

For surjective (onto) ,it's clear that is a polynomial function of odd degree= 3 thus its surjective.

The function for positive values of x will keep increasing in the range thus it will be strictly increasing and similarly for negative values it will be strictly decreasing hence,it an injective function. You can further prove it's injection by

F (x)=f (y) -> x=y

5.a. a~b is an equivalence relation because it satiskies the properties

1. Reflexivity - a~a implies a-a which is zero is an element of z hence satisfying the reflexive property

2. Symmetry- a~b should imply b~a as well. So a-b ie an element of Z and so is b-a which will be an integer hence satisfying this property.

3. a~b ,b~c should imply a~c , if a-b and b-c are element of Z

Then so will a-c which is true because a-b+b-c= a-c will also be an integer.

Thus a~b is a proved equivalence relation.

Since ~ is an equivalence relation on a set Z and 4 is element of Z then set [4]= {x element Z l x-4} is called the equivalence class of Z containing 4.

The members of equivalence class of 4 2/3= 14/3

{14, 28/6,..}

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