OPERATIONS ON SETS

BY: ANGELO CARLON

Union

If A and B be two sets, then the union (A U B) is defined as a set of all those elements which are either in A or in B or in both.

Symbolically, A U B={x: x A v x E B}

Ex. Let A = {a, b, c, d, e} B = {a, e, I, o, u} (A U B) = {a, b, c, d, e, I, o, u}

Intersection

If A and B be two sets, then the intersection (A ∩ B) is defined as a set of all those elements which are common to both the sets.

Symbolically, (A∩B) = {x:xA Ʌ xB}

Ex. A = {a, b, c, d, e} B= {a, e, I, o, u} (A ∩ B) = {a, e}

Difference

If A and B be two sets, then the difference (A-B) is defined as a set of all those elements of A which are not in B,

Symbolically, (A-B) = {x | x A and xB}

Example A = {a, b, c, d, e, f} B = {a, c, I, o, u, k} (A-B) = {b, d, e, f}

Symmetric Difference

If A and B be two sets, then the symmetric difference (A∆B) or (A +o B) is defined as a set of all those elements which are either in A or in B but not in both.

Symbolically, (A +o B) = (A-B) U (B-A)

Ex: Let A = {a, b, c, k, p, q, r, s} B= {b, k, q, m, n, o, t} So, (A-B)= {a, c, p, r, s} and (B-A)= {m, n, o, t} Therefore (A +o B) = (A-B) U (B-A) ={a, c, p, r, s, m, n,

o, t}

Complement of a Set

If A be a set, then the complement of A is given as Ac , A’ or A and is defined as a set of all those elements of the universal set U which are not in A

Symbolically, Ac = {x|xU and xA}

__

Ex. A = {b, c, k, d, I, p, q, r, s, t} So, we can take the universeal set U

= {a,b,c, …, x, y, z} Therefore, Ac = U-A ={a, e, f, g, h, j, l,

m, n, o, u, v, w, x, y, z}

Theorem

Commutative laws: (A U B)=((b U A) (A ∩ B)=(B ∩ A)

Associative laws: A U (B U C)= (A U B) U C A ∩ (B ∩ C)=(A ∩ B) ∩ C

Idempotent laws: A U A = A A ∩ A = A

Identity laws: A U φ = A A ∩ U = A

Bound laws: A U U = U A ∩ φ = φ

Absorption laws: A U (A ∩ B) = A A ∩ (A U B) = A

Complement laws: A U Ac = U A ∩ Ac = φ

Involution law: (Ac )c = A

Distributive laws: A ∩ (B U C)=(A ∩ B) U (A ∩ C) A U (B ∩ C)=(A U B) ∩ (A U C)

de Morgan’s Law

(A U B)c =(Ac ∩ Bc ) (A ∩ B)c =(Ac U Bc )

Disjoint Sets

Two sets A and b are called disjoint or non-overlapping if both sets have no common element.

Mathematically, (A ∩ B)= φ Venn Diagram:

Product of Sets

The product of two sets A and B is the set of all those order pairs whose first coordinate is an element of A and the second coordinate is an element of B denoted by (A x B)

Ex. A = {1,2,3,5} B={4,9,25} So, (A x B)={(1,4), (1,9), (1,25),

(2.4), (2,9), (2,25), (3,4), (3,9), (3,25), (5,4), (4,9), (5,25)}

Fundamental Products

Ex. Consider 3 sets a,

B and C. The fundamental products of these three sets are as follows, which are 2^3 in number.

A ∩ B ∩ C A’ ∩ B ∩ C A ∩ B’ ∩ C A ∩ B ∩ C’ A’ ∩ B’ ∩ C A ∩ B’ ∩ C’ A’ ∩ B ∩ C’ A’ ∩ B’ ∩ C’

End