# Tips and Tricks and Shortcuts for Set Theory

## Shortcuts for Set theory

A set is a collection of some items (elements). To define a set we can simply list all the elements in curly brackets. This page is all about Tips and Tricks for Set Theory.

For Ex- To define a set A that consists of the two elements x and y, we write A={x,y}. To say  that y belongs to A, we write  y ∈ A, where “∈” is pronounced “belong to”. To say that an element does not belong to a set, we use “∉”.

For example – B ∉ A

The set of natural number N = {1,2,3……}. Z={⋯,−3,−2,−1,0,1,2,3,⋯} “> Q

• The set of integers Z={….., -3,-2,-1,0,1,2,3…..}.
• The set of Real number is R
• The set of Rational number is Q

Example of Subsets –
• If E= {1,4} and C = {1,4,9}, then E⊂C
• N⊂Z
• Q⊂Z

### Tips and Tricks For Set Theory

Question 1 – In a presidential election, there are four candidates. Call them A,B,C and D. Based on our polling analysis, we estimate that A has 20% chance of winning the election. While B has 40% chance of winning. What is the probability that a or A or B win the election?

Options
(a)
0.8
(b)
0.6
(c)
0.9
(d)
0.56
(e)
None of these

Explanations

Notice that the events that {A wins}, {B wins}, {C wins} and {D wins} are disjoint since more than one of them cannot occur at the same time. If A wins then B cannot win. From the third axiom of probability, the probability of the union of two disjoint events is the summation of individual probabilities. Therefore,

P(A wins or B wins) = P( {A wins} ∪ { B wins} )

=  P ( {A wins} + P{B wins} )

= 0.2 + 0.4
= 0.6
Correct options (B)

Question 2 – If you roll a fair die. What is the probability of E = {1,5}?

Options
(a)
$\frac{1}{3} \$

(b) $\frac{2}{8} \$

(c) $\frac{2}{6} \$

P({1}) = P({2}) =…….= P({6})

1 = P{S}
= P ( {1}∪{2}∪ ….. ∪{6} )

= P({1}) + P({2}) + ……. +P({6})
= 6P({1})

Thus,

P({1}) = P({2}) = …. = P({6}) = $\frac{1}{6} \$

P(E) = P({1,5}) = P({1}) + P({5}) = $\frac{2}{6} \$ = $\frac{1}{3} \$

Correct options (A)