De Morgan's Law, Combined Set Operations And Properties Of Set

De Morgan’s Theorem gives the following equations on set operations.

(A ∪ B)’ = A’ ∩ B’

(A ∩ B)’ = A’ ∪ B’

De Morgan’s Theorem can be used to simplify expressions that involve set operations. It is as well used in Physics for the simplification of Boolean expressions and digital circuits.

SETS and De Morgan's Law-A

AU (BC) = ( AUB) ∩ (AUC): Union of Sets is distributive over intersection of sets.

A∩ (BUC) = (A∩B) U (A∩C): Intersection of sets is distributive over union of sets.

Demorgan’s Laws-B:

(a) (AUB)I = AI ∩ BI

The complement of union of sets is the intersection of their complements

b) (A∩B)I = AI ∪ BI

The complement of intersection of intersection of sets is the union of their complements

n (A) represents the number of elements in set A.

n (AUB) = n (A) + n (B) - n (A∩B)

n (A∩B) = n (A) + n (B) – n (AUB)

Note: (1) If P and Q are disjoint sets: P∩Q = ø, P - Q = P, Q-P = Q

2) If A and B are equal sets, then

AUB = A or B

A∩B = A or B A – B = ø B – A = ø

3) If B is a proper subset of A [B is a non zero set] then AUB = A, A∩B = B, B - A = ø

4) If B is a null set or empty set and A is an other set, then AUB = A, A∩B = B, A-B=A, B-A=B

If A and B are disjoint sets [No elements is common]: A∩B = ø, n(A∩B) = 0 and n(AUB) = n(A) +n(B)

DEMORGAN’S LAWS-C:

(AUB)l = Al ∩ Bl represents the complement of union of sets in the intersection of their complements.

(A∩B)l = AIUBI represents the complement of intersection of sets is the union of their complements.

A = {Prime numbers < 12}, i.e. A= {2, 3, 5, 7, 11}

B ={x/x Є N 2 ≤ 5}, i. e. B = {2, 3, 4, 5}

C = {Perfect square number < 16}, i.e. C = {1, 4, 9}

In Disjoint sets: A∩B = {} or ∅ and n (A∩B) = 0

A set is a collection of well defined objects.

· A set can be represented in two ways: they are Rule Method and Roster Method.

Ex. Rule Method

A={x/x is a Prime Number < 8}

Roster Method

B= {2,3,5,7}

Types of sets

a) Finite Set: A set which has countable number of elements

b) Infinite Set: A set in which the number of elements cannot be counted

c) Singleton Set: A set which contains only one element

d) Null Set (Empty Set): A set with no element in it

Properties of set

Commutative Property: (a)

AUB = BUA →Union

b) A∩B = B∩A → Intersection

Distributive Property: (a) AU (B∩C) = (AUB) ∩ (AUC)

Union of sets is distributive over intersection of sets

Commutative Laws

The "Commutative Laws" say that we can swap numbers over and still obtain the same answer when we add:

a + b = b + a

or when we multiply:

a × b = b × a

Associative Laws

The "Associative Laws" say that it doesn't matter how we group the numbers (i.e. which we calculate first) when we add as shown below:

(a + b) + c = a + (b + c)

... or when we multiply:

(a × b) × c = a × (b × c)

Sometimes it is easier to add or multiply in a different order:

What is 19 + 36 + 4?

19 + 36 + 4 = 19 + (36 + 4) = 19 + 40 = 59

Or to rearrange a little:

What is 2 × 16 × 5?

2 × 16 × 5 = (2 × 5) × 16 = 10 × 16 = 160

Distributive Law

The "Distributive Law" is the BEST one of all, but requires careful attention.

And we write it as follows:

a × (b + c) = a × b + a × c

Try the calculations yourself:

3 × (2 + 4) = 3 × 6 = 18

3 × 2 + 3 × 4 = 6 + 12 = 18

Either way leads you to the same answer.

In English we can say:

We get the same answer when we:

multiply a number by a group of numbers added together, or

do each multiplication separately then add them

Uses:

Every so often it is easier to break up a difficult multiplication:

Example: What is 6 × 204 ?

6 × 204 = 6×200 + 6×4 = 1,200 + 24 = 1,224

Or to combine:

Example: What is 16 × 6 + 16 × 4?

16 × 6 + 16 × 4 = 16 × (6+4) = 16 × 10 = 160

We can as well make use of it in subtraction:

Example: 26 × 3 – 24 × 3

26 × 3 – 24 × 3 = (26 - 24) × 3 = 2 × 3 = 6

We could as well use it for a long list of additions:

Example: 6 × 7 + 2 × 7 + 3 × 7 + 5 × 7 + 4 × 7

6 × 7 + 2 × 7 + 3 × 7 + 5 × 7 + 4 × 7 = (6 + 2 + 3 + 5 + 4) × 7 = 20 × 7 = 140

And those are the Laws it is essential to note that

The Commutative Law does not work for division:

Example:

12 / 3 = 4, but

3 / 12 = ¼

 Again, the Associative Law does not work for subtraction:

Example:

(9 – 4) – 3 = 5 – 3 = 2, but

9 – (4 – 3) = 9 – 1 = 8

Furthermore, the Distributive Law does not work for division:

Example:

24 / (4 + 8) = 24 / 12 = 2, but

24 / 4 + 24 / 8 = 6 + 3 = 9

Set Theory: Universal Set

In this section, we will discover what a Universal set is and how it may be represented in a Venn Diagram.

A universal set is the set of all elements under consideration, denoted by capital U or occasionally capital E.

Example:

Given that U = {5, 6, 7, 8, 9, 10, 11, 12}, list the elements of the following sets.

a) A = {x : x is a factor of 60}

b) B = {x : x is a prime number}

Solution:

The elements of sets A and B can only be selected from the given universal set U .

a) A = {5, 6, 10, 12}

b) B = {5, 7, 11}

Combined Operations

Combined operations involve the intersection, union and complement of sets. The procedure for calculating this type is: carry out the operations within brackets first. Other operations are then carried out starting from left to right.

Example:

Given that U = {x : 1 ≤ x ≤ 10, x is an integer},

G = {x : x is a prime number},

H = {x : x is an even number},

P = {1, 2, 3, 4, 5}.

List the elements of:

a) G ∩ H ∪ P

b) (G ∩ P) ’ ∪ H

c) H ’ ∩ (G ∪ P )

d) (P ∪ H ∪ G) ’ ∩ (G ∩ H)

Answer

G = {2, 3, 5, 7}, H = {2, 4, 6, 8, 10}

a) G ∩ H ∪ P = {2} ∪ P ← G ∩ H = {2}

= {1, 2, 3, 4, 5}

b) (G ∩ P) ’ ∪ H = {1, 4, 6, 7, 8, 9, 10} ∪ H

= {1, 2, 4, 6, 7, 8, 9, 10}

c) H ’ ∩ (G ∪ P ) = H ’ ∩ {1, 2, 3, 4, 5, 7}

= {1, 3, 5, 7}

d) (P ∪ H ∪ G) ’ ∩ (G ∩ H) = {9} ∩ (G ∩ H)

= {9} ∩ {2} = { }

A few questions to test your ability

1. Union of sets is distributive over intersection of sets can be represented as

a) AUB = BUA

b) (AUB) UC = AU (BUC)

c) AU (B∩C) = (AUB) ∩ (AUC)

d) A∩ (BUC) = (A∩B) U (A∩C)

2. If P = {1, 2, 3, 4} and Q= {3, 4, 5} then P-Q is

a) {0, 1, 2}

(b) {1, 6}

(c) {1, 2}

(d) {3, 4}

3. A is a subset of ‘U’. If n (U) = 12 and n (AI) = 7, then n (A) is

a) 19

(b) 12

(c) 7

(d) 5

4. Out of 1000 students 750 play cricket, 350 play volley ball, 150 play both the games. Number of students who do not play any of these 2 games is

a) 500 b) 250 c) 200 d) 50

5. If A = {a, b, c, d, e}, B={ a, c, e, g, h} and C={c ,f, g} then A ∩ (BUC) =

a) {c} b) {b, d} c) {e, g} d) {a, c, e}

6. If A={1, 2, 3, 4}, B={2, 3, 5} then n(A∩B) =

a) 2 b) 3 c) 4 d) 7

7. U = {2, 3, 5, 6, 10}, A= {5,6} then diagram which represents AI is

8. If A = {x N/1 ≤ x ≤ 4}, and B= {3, 4, 5} then A ∩ B is

a) {1, 2, 3, 4, 5} b) {1, 2, 3, 4, 5} c) {1, 2} d) {3, 4}

9. If (AUB)I = {2, 4, 6}, then AI ∩ BI is equal to

a) {1, 2, 3, 4, 5, 6} b) {2, 4, 6} c) {1, 3, 5} d) { } View more on psalmfresh.blogspot.com for more educating and tricks article Thank you