University:
East Georgia State College
Course:
MATH 1540 | Calculus I
Academic year:

2024
Views:

274

Pages:

4
Author:

somospumasl3o

Question 1 Define discrete math? Answer: Step 1 Definition Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. Discrete mathematics includes the following main subjects: Permutations, combinatorial, number theory, logic, graph theory, and probability. Step 2 Explanation The set of objects studied in discrete mathematics can be finite or infinite. The term finite mathematics is sometimes applied to parts of the field of discrete mathematics that deal with finite sets, particularly those areas relevant to business. Question 2 Determine whether each statement is true or false. For each true statement, give a direct evidence as justification. For each false statement, give a counterexample as defense (separate calculation): If 𝑥 + 𝑦 x+y is an even integer, then 𝑥 x and 𝑦 y are both even integers. If 𝑥 2 = 𝑦 2 x 2 =y 2 , then 𝑥 = 𝑦 x=y. The mean of two even numbers is even. If 𝑥 x and 𝑦 y are even integers, then 𝑥 + 𝑦 x+y is an even integer. Answer 1: (i) 𝑥 + 𝑦 x+y is an even integer. Let us take 𝑥 = 5 x=5 and 𝑦 = 3 y=3. Both 𝑥 + 𝑦 x+y are odd but 𝑥 + 𝑦 = 5 + 3 = 8 x+y=5+3=8. 2 − 4 2−4 is even. Hence, (i) is false. (ii) 𝑥 2 = 𝑦 2 x 2 =y 2 . Let us take 𝑥 = 2 x=2 and 𝑦 = 2 y=2. Now, 𝑥 2 = ( − 2 ) 2 = 4 = 𝑦 2 x 2 =(−2) 2 =4=y 2 , but 𝑥 ≠ 𝑦 x  =y. Hence, (ii) is not true. (iii) Mean of two 𝑥 = 6 x=6 and 𝑦 = 4 y=4. 𝑥 + 𝑦 2 = 6 + 4 2 = 5 2 x+y = 2 6+4 =5 = 2 ( 5 + 1 ) =2(5+1) The mean of an even integer is odd. Hence, (iii) is not true. (iv) 𝑥 x and 𝑦 y are even integers. Let 𝑥 x and 𝑛 + 𝑦 n+y integers even. 𝑆 𝑒 𝑡 𝑛 = 2 𝑠 ℎ 𝑒 𝑛 𝑐 𝑒 < = + 𝑦 𝑒 𝑣 𝑒 𝑛 ℎ 𝑒 𝑟 𝑒 𝑖 𝑛 𝑡 𝑒 𝑔 𝑒 𝑟 𝑠 . Setn=2shence<=+yevenhereintegers. Hence, (x) Answer 2 (Continued): Solution: If 𝑥 + 𝑦 x+y is an even integer, then 𝑥 x and 𝑦 y are both even integers. This statement is false. A counterexample is 𝑥 = 3 x=3 and 𝑦 = 5 y=5, where 𝑥 + 𝑦 = 8 x+y=8 is an even integer, but neither 𝑥 x nor 𝑦 y is even. If 𝑥 2 = 𝑦 2 x 2 =y 2 , then 𝑥 = 𝑦 x=y. This statement is false. A counterexample is 𝑥 = − 2 x=−2 and 𝑦 = 2 y=2, where 𝑥 2 = 𝑦 2 = 4 x 2 =y 2 =4, but 𝑥 ≠ 𝑦 x  =y. The mean of two even numbers is even. This statement is true. Let 𝑥 = 2 𝑎 x=2a and 𝑦 = 2 𝑏 y=2b for some integers 𝑎 a and 𝑏 b. Then, the mean of 𝑥 x and 𝑦 y is: 𝑥 + 𝑦 2 = 2 𝑎 + 2 𝑏 2 = 𝑎 + 𝑏 2 x+y = 2 2a+2b =a+b Since 𝑎 a and 𝑏 b are integers, 𝑎 + 𝑏 a+b is an integer. Therefore, the mean of two even numbers is an integer, and thus even. If 𝑥 x and 𝑦 y are even integers, then 𝑥 + 𝑦 x+y is an even integer. This statement is true. Let 𝑥 = 2 𝑎 x=2a and 𝑦 = 2 𝑏 y=2b for some integers 𝑎 a and 𝑏 b. Then: 𝑥 + 𝑦 = 2 𝑎 + 2 𝑏 = 2 ( 𝑎 + 𝑏 ) x+y=2a+2b=2(a+b) Since 𝑎 + 𝑏 a+b is an integer, 𝑥 + 𝑦 x+y is a multiple of 2 2, and hence even. Question 3 Give full and correct answer in how many ways can a set of two positive integers less than 100 be chosen? Answer: For a set of two integers 𝑎 , 𝑏 a,b (where 𝑎 < 𝑏 < 100 a