**Find union of A and B and represent it using venn diagram.**

**(i) A = {1, 2, 3, 4, 8, 9}, B = {1, 2, 3, 5} **

**(ii) A = {1, 2, 3, 4, 5}, B = {4, 5, 7, 9} **

**(iii) A = {1,2,3}, B = {4, 5, 6} **

**(iv) A = {1, 2, 3, ,4 ,5}, B = {1, 3, 5} **

**(v) A = {a, b, c, d}, B = {b, d, e, f} **

Solution:

(i) A U B = {1, 2, 3, 4, 5, 8, 9}

(ii) A U B = {1, 2, 3, 4, 5, 7, 9}

(iii) A U B = {1, 2, 3, 4, 5, 6}

(iv) A U B = {1, 2, 3, 4, 5}

(v) A U B = {a, b, c, d, e, f}

**Find the intersection of A and B, and respect it by Venn diagram:**

**(i) A = {a, c, d, e}, B = {b, d, e, f} **

**(ii) A = {1, 2, 4, 5}, B = {2, 5, 7, 9} **

**(iii) A = {1, 3, 5, 7}, B = {2, 5, 7, 10, 12} **

**(iv) A = {1, 2, 3}, B = {5, 4, 7} **

**(v) A = {a, b, c}, B = {1, 2, 9}**

Solution:

(i) A ∩ B = {d, e}

(ii) A ∩ B = {2, 5}

(iii) A ∩ B = {5, 7}

(iv) A ∩ B = { }

**Find A**∩**B and A**∩**B when:**

**(i) A is the set of all prime numbers and B is the set of all composite natural numbers: **

**(ii) A is the set of all positive real numbers and B is the set of all negative real numbers: **

**(iii) A = N and B = Z: **

**(iv) A = {x /x **∩ **Z and x is divisible by 6} and **

**B = {x / x **∩ **Z and x is divisible by 15} **

**(v) A is the set of all points in the plane with integer coordinate and B is the set of all points with rational coordinates **

Solution:

(i) A = {2, 3, 5, 7….}

B = {1, 4, 6….}

A U B = {1, 2, 3, 4} = N

A ∩ B = { }

(ii) A = R+

B = R+

A U B = R – {10}

i.e. A U B ={set of non zero real numbers}

A U B = { }

(iii) A = N B = N

A U B = Z

A ∩ B = N

(iv) A U B = {x / x € Z and x is divisible by 6 and 15} and

A ∩ B = {x / x € Z and x is divisible by 30}

[LCM of 6 and 15 = 30]

(v) A U B = the set of all points with rational co-ordinates = B.

A ∩ B = the set of all points with rational co-ordinates = A.

**Give examples to show that**

**(i) A **U **A = A and A **∩ **A = A **

**(ii) If A **⸦ **B, then A **U **B = B and A **∩ **B =A. can you prove these statements formally? **

Solution:

(i) If A = {2 4 6 8}

Then A U A = {2, 4, 6, 8……} = A

A ∩ A = {2, 4, 6, 8…….} = A

Hence A U A = A and A ∩ A = A

(ii) A = {1, 3, 5, 7, 9……}

B = {1, 2, 3, 4, 5……}

We see that A ⸦ B

A U B = {1 2 3 4…..} = B

A ∩ B = B

A ∩ B = {1, 3, 5……} = A

A ∩ B = A

**What is A**U Φ**and A**∩ Φ**for a set A?**

**Solution:**

** **AUΦ = A ; A ∩Φ = Φ

good exercices . thanks