Define a function
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Discrete Mathematics With Applications
- Prove that if f is a permutation on A, then (f1)1=f.arrow_forwardLabel each of the following statements as either true or false. We say that cF is a solution to the polynomial equation f(x)=0 if and only if f(c)=0inF.arrow_forwardLet T be a linear transformation from P2 into P2 such that T(1)=x,T(x)=1+xandT(x2)=1+x+x2. Find T(26x+x2).arrow_forward
- Show that if ax2+bx+c=0 for all x, then a=b=c=0.arrow_forwardA relation R on a nonempty set A is called asymmetric if, for x and y in A, xRy implies yRx. Which of the relations in Exercise 2 areasymmetric? In each of the following parts, a relation R is defined on the set of all integers. Determine in each case whether or not R is reflexive, symmetric, or transitive. Justify your answers. a. xRy if and only if x=2y. b. xRy if and only if x=y. c. xRy if and only if y=xk for some k in . d. xRy if and only if xy. e. xRy if and only if xy. f. xRy if and only if x=|y|. g. xRy if and only if |x||y+1|. h. xRy if and only if xy i. xRy if and only if xy j. xRy if and only if |xy|=1. k. xRy if and only if |xy|1.arrow_forwardWrite each of the following polynomials as a products of its leading coefficient and a finite number of monic irreducible polynomials over 5. State their zeros and the multiplicity of each zero. 2x3+1 3x3+2x2+x+2 3x3+x2+2x+4 2x3+4x2+3x+1 2x4+x3+3x+2 3x4+3x3+x+3 x4+x3+x2+2x+3 x4+x3+2x2+3x+2 x4+2x3+3x+4 x5+x4+3x3+2x2+4xarrow_forward
- For each of the following pairs and decide whether is onto or one-to-one and justify all negative answers.arrow_forward6х + 5 Define a functionf : R* → R as follows: for all real numbers r, f(x) = Then f is both one-to-one and onto.arrow_forwardA function f:R →→R, where R is the set of real- ax² + 5x - 8 numbers, is defined by f(x) = Find a + 5x8x² the interval of values of a for which is onto. Is the function one to one for a = 3? Justify your answer.arrow_forward
- Let a, b, c, d be real numbers where c and d are not both zero. Let X = {x is a real number | cx + d =/= 0}. The function f : X -> R is defined by f(x) = (ac + b)/(cx + d) a. Under what condition/s on a, b, c, d will f be one-to-one? b. In addition to the condition/s above, justify that f is a bijection iff c = 0.arrow_forwardLet andarrow_forwardSuppose the function f has the property that there exists a number B such that |S(x) – f(c)| < B|x – c| for all x in the interval (c – p, c+p). Prove that f is con- tinuous at c.arrow_forward
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