Let X and Y be non-empty sets. If A1 and A2 are subsets of X, and B1 and B2 subsets of Y. Show that (A1×B1)U(A2×B2) = (A1UA2)×(B1UB2).

Topology Sets and Functions (XLIV) Functions Let X and Y be non-empty sets. If A1 and A2 are subsets of X and B1and B2 subsets of Y, show that (A1×B1)U(A2×B2) = (A1UA2)×(B1UB2). See the attached file.

Let f:X→Y be an arbitrary mapping. Define a relation in X as follows: x1 ~ x2 means that f(x1) = f(x2). Show that this is an equivalence relation and describe the equivalence sets.

Topology Sets and Functions (XLV) Functions Let f:X→Y be an arbitrary mapping. Define a relation in X as follows: x1 ~ x2 means that f(x1) = f(x2). Show that this is an equivalence relation and describe the equivalence sets. See the attached file.

In the set R of all real numbers, let x ~ y means that x – y is an integer. Show that this is an equivalence relation and describe the equivalence sets.

Topology Sets and Functions (XLVI) Functions In the set R of all real numbers, let x ~ y means that x – y is an integer. Show that this is an equivalence relation and describe the equivalence sets. See the attached file.

Let I be the set of all integers and let m be a fixed positive integer. Two integers a and b are said to be congruent modulo m-symbolized by a ≡ b (mod m) - if a – b is exactly divisible by m, i.e., if a – b is an integral multiple of m. Show that this is an equivalence relation , describe the equivalence set, and state the number of distinct equivalence sets.

Topology Sets and Functions (XLVII) Functions Let I be the set of all integers and let m be a fixed positive integer. Two integers a and b are said to be congruent modulo m-symbolized by a ≡ b (mod m) - if a – b is exactly divisible by m, i.e., if a – b is an integral multiple of m. Show that this is an equivalence relation , describe the equivalence set, and state the number of distinct equivalence sets. See the attached file.

Decide which ones of the three properties of reflexivity, symmetry, and transitivity are true for each of the following relations in the set of all positive integers: m ≤ n, m < n, m divides n. Are any of these equivalence relations?

Topology Sets and Functions (XLVIII) Functions Decide which ones of the three properties of reflexivity, symmetry, and transitivity are true for each of the following relations in the set of all positive integers: m ≤ n, m < n, m divides n. Are any of these equivalence relations? See the attached file.