De Morgan's Laws | Venn Diagrams | Proofs Maths | Sets

De Morgan's Laws | Venn Diagrams | Proofs Maths | Sets

          De Morgan’s father (a British national) was in the service of East India Company, India. Augustus De Morgan (1806-1871) was born in Madurai, Tamilnadu, India. His family moved to England when he was seven months old. He had his education at Trinity college, Cambridge, England. De Morgan’s laws relate the three basic set operations Union, Intersection and Complementation.

Venn diagram, graphical method of representing categorical propositions and testing the validity of categorical syllogisms, devised by the English logician and philosopher John Venn (1834–1923). Long recognized for their pedagogical value, Venn diagrams have been a standard part of the curriculum of introductory logic since the mid-20th century.

Venn introduced the diagrams that bear his name as a means of representing relations of inclusion and exclusion between classes, or sets. Venn diagrams consist of two or three intersecting circles, each representing a class and each labeled with an uppercase letter. Lowercase x’s and shading are used to indicate the existence and nonexistence, respectively, of some (at least one) member of a given class.

Two-circle Venn diagrams are used to represent categorical propositions, whose logical relations were first studied systematically by Aristotle. Such propositions consist of two terms, or class nouns, called the subject (S) and the predicate (P); the quantifier all, no, or some; and the copula are or are not. The proposition “All S are P,” called the universal affirmative, is represented by shading the part of the circle labeled S that does not intersect the circle labeled P, indicating that there is nothing that is an S that is not also a P. “No S are P,” the universal negative, is represented by shading the intersection of S and P; “Some S are P,” the particular affirmative, is represented by placing an x in the intersection of S and P; and “Some S are not P,” the particular negative, is represented by placing an x in the part of S that does not intersect P.

What is an Equation

An equation says that two things are equal. It will have an equals sign "=" like this:

x

+

2

=

6

That equation says: what is on the left (x + 2) is equal to what is on the right (6)

So an equation is like a statement "this equals that"

Parts of an Equation

So people can talk about equations, there are names for different parts (better than saying "that thingy there"!)

Here we have an equation that says 4x − 7 equals 5, and all its parts:

A Variable is a symbol for a number we don't know yet. It is usually a letter like x or y.

A number on its own is called a Constant.

A Coefficient is a number used to multiply a variable (4x means 4 times x, so 4 is a coefficient)

Variables on their own (without a number next to them) actually have a coefficient of 1 (x is really 1x)

Sometimes a coefficient is a letter like a or b instead of a number:

Example: ax² + bx + c

• x is a variable
• a and b are coefficients
• c is a constant

An Operator is a symbol (such as +, ×, etc) that shows an operation (ie we want to do something with the values).

A Term is either a single number or a variable, or numbers and variables multiplied together.

An Expression is a group of terms (the terms are separated by + or − signs)

So, now we can say things like "that expression has only two terms", or "the second term is a constant", or even "are you sure the coefficient is really 4?"

Exponents

The exponent (such as the 2 in x2) says how many times to use the value in a multiplication. Examples: 82 = 8 × 8 = 64 y3 = y × y × y y2z = y × y × z

Exponents make it easier to write and use many multiplications

Example: y⁴z² is easier than y × y × y × y × z × z, or even yyyyzz

Polynomial

Example of a Polynomial: 3x² + x - 2

A polynomial can have constants, variables and the exponents 0,1,2,3,...

But it never has division by a variable.