practice questions
... (a) If x is even, then x2 is even. (b) x is even implies x2 is even. (c) x is even only if x2 is even. (d) It is necessary that x2 be even for x to be even. Note that “x2 is even is necessary for x is even” is not even a correct English sentence! Part of the point of this question was to make people ...
... (a) If x is even, then x2 is even. (b) x is even implies x2 is even. (c) x is even only if x2 is even. (d) It is necessary that x2 be even for x to be even. Note that “x2 is even is necessary for x is even” is not even a correct English sentence! Part of the point of this question was to make people ...
Version 1.0 of the Math 135 course notes - CEMC
... Showing Two Sets Are Equal . . . . . . . . . . . . . 8.3.1 Converse of an Implication . . . . . . . . . . . 8.3.2 If and Only If Statements . . . . . . . . . . . 8.3.3 Set Equality and If and Only If Statements . ...
... Showing Two Sets Are Equal . . . . . . . . . . . . . 8.3.1 Converse of an Implication . . . . . . . . . . . 8.3.2 If and Only If Statements . . . . . . . . . . . 8.3.3 Set Equality and If and Only If Statements . ...
Chapter 5A - Polynomial Functions
... Now if we examine each of the terms in the second factor we see that as x gets large either positively or negatively every one of the quotients must get smaller and smaller. That is every p term which of the form n−ii goes to zero as long as the exponent n − i is positive. So, for large x x the seco ...
... Now if we examine each of the terms in the second factor we see that as x gets large either positively or negatively every one of the quotients must get smaller and smaller. That is every p term which of the form n−ii goes to zero as long as the exponent n − i is positive. So, for large x x the seco ...
of integers satisfying a linear recursion relation
... the r residues A 0, ■ ■• , AT_Xare said to form a cycle (-4) belonging to F{x). Two such cycles are said to be equal if either can be obtained from the other by a cyclic permutation of its elements. Let L be any residue. If the cycle LA a, • • ■, LAT^Xequals the cycle (4), then L is called a multipl ...
... the r residues A 0, ■ ■• , AT_Xare said to form a cycle (-4) belonging to F{x). Two such cycles are said to be equal if either can be obtained from the other by a cyclic permutation of its elements. Let L be any residue. If the cycle LA a, • • ■, LAT^Xequals the cycle (4), then L is called a multipl ...
34(5)
... If ^ ( x , j , z,...) is a polynomial, the difference ^(x, y, z, ...)-$(y, x, z,...) is divisible by x - y. Following Newton, for any pair of variables (x, y), one defines a divided difference operator, d acting on the ring of polynomials as ^ ( x , j ; , . . . ) - > ^ ( x , j ; , . . . ) ^ r v ?^? ...
... If ^ ( x , j , z,...) is a polynomial, the difference ^(x, y, z, ...)-$(y, x, z,...) is divisible by x - y. Following Newton, for any pair of variables (x, y), one defines a divided difference operator, d acting on the ring of polynomials as ^ ( x , j ; , . . . ) - > ^ ( x , j ; , . . . ) ^ r v ?^? ...
