Mathematical Inclusion and Exclusion can be stated as The Law of Non-Contradiction.
“If A, then not B.”
“If this, then no other answer is possible.”
For instance, 1+1=2; 2+2=4; 3+3=6; 4+4=8; 5+5=10. 2X will always be equal to the number added to itself. One plus one cannot equal to 10, not even in base two, which is the foundation for a computer’s bits and bytes.
When we insert some number for the “X”, then an even number will be the result. Even numbers are included in the formula 2X=Y. Odd numbers are excluded.
In mathematics as in life in general, there is one right answer to every equation.
In algebra, the knowns of the equation limit the unknown of the same. 5+a=25, then a=25-5, and a=5.
This is pretty straight-forward. In the previous equation, the unknown cannot equal to any other number than five or the equation will be incorrect.
In other words, five is the all-inclusive answer to the algebraic equation 5+a=25. The other integers are excluded from this equation as being possible answers.
To find the circumference of a right triangle one must use the Pythagorean Theorem. The square of the hypotenuse is equal to the sums of the squares of the two sides. axa + bxb = cxc or A squared plus B squared equals C squared.
For instance, 6×6 + 8×8 = CxC; 36+64=CxC; 100=CxC; 10=C. In this particular right triangle, side A = 6; side B = 8; and side C =10. Ten is the inclusive answer for C. All other answers are excluded.
Geometric shapes demonstrated the inclusivity/exclusivity principle.
Triangles are always three-sided, since “tri” as a prefix always means “three.” Shapes with more sides are excluded from this designation.
Squares, parallelograms, rhombuses, and trapezoids are always four-sided figures. Only four-sided shapes are included in these designations. Shapes with more or less sides are excluded.
Pentagons have five sides. “Pent” as a prefix always means “five” in mathematics. Only five-sided shapes are included in this designation. Shapes with more or less sides are excluded. (At least one famous Pentagon, also, includes military personnel.)
Hexagons have six sides. “Hex” as a prefix always means “six” in mathematics. Only six-sided shapes are included in this designation. Shapes with more or less sides are excluded.
Mathematics have the Inclusion/Exclusion Principle in common with life in general. The characteristics of an area of life that maintains successful function are included. Those characteristics that are non-functional are excluded.
In education, when teachers educate their students in accordance with life as it really is, then their students are able to function in life. Education is inclusive with truth. Education is exclusive with error.
In horticulture, sun-loving plants, like the Bradford Pear Tree, grow tall and strong, when included in an environment of full-sun. When a Bradford Pear Tree is excluded from full-sun in partial shade or full shade as an understudy tree stunted growth is the result. When shade-loving plants, like Boston Ferns, are given a protected environment of shade all day long, the the fern thrives as is inclusive in its nature. Excluded from shade in full-sun, these ferns can quickly turn brown and die.
In sports, when the actions of players are included within the boundaries of the rules, the team has the greater potential for victory than the team has when their actions are excluded from the boundaries of the rules.
Who wants to play with a kid, who is always changing the boundaries to help himself win? Such an individual is quickly excluded from our circle of friends.
