Writing Proofs About Sets

Prof. Susan Older

22 September 2016

(CIS 375) Proof Templates (Sets) 22 Sept 2016 1 / 5

Proving Existential Statements Directly

To prove a statement of the form 9x 2 U,P(x):

1 Give a specific value v 2 U

2 Show that P(v) is true.

Claim: There is a set X such that {1, 2} ✓ X and |2X | > 10.

Proof: (direct)Let X = {1, 2, 3, 4}. Clearly {1, 2} ✓ X .

Recall that, for any set Y , |2Y | = 2|Y |. Because |X | = 4, we can see that

|2X | = 2|X | = 24 = 16 > 10.

Therefore, the claim is true.

(CIS 375) Proof Templates (Sets) 22 Sept 2016 2 / 5

Proving Existential Statements Directly

To prove a statement of the form 9x 2 U,P(x):

1 Give a specific value v 2 U

2 Show that P(v) is true.

Claim: There is a set X such that {1, 2} ✓ X and |2X | > 10.

Proof: (direct)Let X = {1, 2, 3, 4}. Clearly {1, 2} ✓ X .

Recall that, for any set Y , |2Y | = 2|Y |. Because |X | = 4, we can see that

|2X | = 2|X | = 24 = 16 > 10.

Therefore, the claim is true.

(CIS 375) Proof Templates (Sets) 22 Sept 2016 2 / 5

Proving Existential Statements Directly

To prove a statement of the form 9x 2 U,P(x):

1 Give a specific value v 2 U

2 Show that P(v) is true.

Claim: There is a set X such that {1, 2} ✓ X and |2X | > 10.

Proof: (direct)Let X = {1, 2, 3, 4}.

Clearly {1, 2} ✓ X .

Recall that, for any set Y , |2Y | = 2|Y |. Because |X | = 4, we can see that

|2X | = 2|X | = 24 = 16 > 10.

Therefore, the claim is true.

(CIS 375) Proof Templates (Sets) 22 Sept 2016 2 / 5

Proving Existential Statements Directly

To prove a statement of the form 9x 2 U,P(x):

1 Give a specific value v 2 U

2 Show that P(v) is true.

Claim: There is a set X such that {1, 2} ✓ X and |2X | > 10.

Proof: (direct)Let X = {1, 2, 3, 4}. Clearly {1, 2} ✓ X .

Recall that, for any set Y , |2Y | = 2|Y |. Because |X | = 4, we can see that

|2X | = 2|X | = 24 = 16 > 10.

Therefore, the claim is true.

(CIS 375) Proof Templates (Sets) 22 Sept 2016 2 / 5

Proving Existential Statements Directly

To prove a statement of the form 9x 2 U,P(x):

1 Give a specific value v 2 U

2 Show that P(v) is true.

Claim: There is a set X such that {1, 2} ✓ X and |2X | > 10.

Proof: (direct)Let X = {1, 2, 3, 4}. Clearly {1, 2} ✓ X .

Recall that, for any set Y , |2Y | = 2|Y |.

Because |X | = 4, we can see that

|2X | = 2|X | = 24 = 16 > 10.

Therefore, the claim is true.

(CIS 375) Proof Templates (Sets) 22 Sept 2016 2 / 5

Proving Existential Statements Directly

To prove a statement of the form 9x 2 U,P(x):

1 Give a specific value v 2 U

2 Show that P(v) is true.

Claim: There is a set X such that {1, 2} ✓ X and |2X | > 10.

Proof: (direct)Let X = {1, 2, 3, 4}. Clearly {1, 2} ✓ X .

Recall that, for any set Y , |2Y | = 2|Y |. Because |X | = 4, we can see that

|2X | = 2|X | = 24 = 16 > 10.

Therefore, the claim is true.

(CIS 375) Proof Templates (Sets) 22 Sept 2016 2 / 5

Proving Existential Statements Directly

To prove a statement of the form 9x 2 U,P(x):

1 Give a specific value v 2 U

2 Show that P(v) is true.

Claim: There is a set X such that {1, 2} ✓ X and |2X | > 10.

Proof: (direct)Let X = {1, 2, 3, 4}. Clearly {1, 2} ✓ X .

Recall that, for any set Y , |2Y | = 2|Y |. Because |X | = 4, we can see that

|2X | = 2|X | = 24 = 16 > 10.

Therefore, the claim is true.

(CIS 375) Proof Templates (Sets) 22 Sept 2016 2 / 5

Proving Existential Statements Directly

To prove a statement of the form 9x 2 U,P(x):

1 Give a specific value v 2 U

2 Show that P(v) is true.

Claim: There is a set X such that {1, 2} ✓ X and |2X | > 10.

Proof: (direct)Let X = {1, 2, 3, 4}. Clearly {1, 2} ✓ X .

Recall that, for any set Y , |2Y | = 2|Y |. Because |X | = 4, we can see that

|2X | = 2|X | = 24 = 16 > 10.

Therefore, the claim is true.

(CIS 375) Proof Templates (Sets) 22 Sept 2016 2 / 5

Proving Universal Statements Directly

To prove a statement of the form 8x 2 U,P(x):

1 Consider an arbitrary (i.e., nonspecific) object a 2 U.

2 Show that P(a) is true.

Claim: Let E = {n 2 Z : n is even}. For all m 2 E , m2

is even.

Proof: (direct)Consider an arbitrary p 2 E . [Need to show: p2 is even, which meansthere exists integer k such that p2 = 2k .]

Because p 2 E , we know p 2 Z and p is even, and hence there is aninteger ` such that p = 2`. It follows that

p

2 = (2`)2 = 4`2 = 2 · (2`2).

Because ` is an integer, so is 2`2. Thus, there is an integer k (namely 2`2)such that p2 = 2k , and p

2 is even. Because p was arbitrary, the claim istrue.

(CIS 375) Proof Templates (Sets) 22 Sept 2016 3 / 5

Proving Universal Statements Directly

To prove a statement of the form 8x 2 U,P(x):

1 Consider an arbitrary (i.e., nonspecific) object a 2 U.

2 Show that P(a) is true.

Claim: Let E = {n 2 Z : n is even}. For all m 2 E , m2

is even.

Proof: (direct)Consider an arbitrary p 2 E . [Need to show: p2 is even, which meansthere exists integer k such that p2 = 2k .]

Because p 2 E , we know p 2 Z and p is even, and hence there is aninteger ` such that p = 2`. It follows that

p

2 = (2`)2 = 4`2 = 2 · (2`2).

Because ` is an integer, so is 2`2. Thus, there is an integer k (namely 2`2)such that p2 = 2k , and p

2 is even. Because p was arbitrary, the claim istrue.

(CIS 375) Proof Templates (Sets) 22 Sept 2016 3 / 5

Proving Universal Statements Directly

To prove a statement of the form 8x 2 U,P(x):

1 Consider an arbitrary (i.e., nonspecific) object a 2 U.

2 Show that P(a) is true.

Claim: Let E = {n 2 Z : n is even}. For all m 2 E , m2

is even.

Proof: (direct)Consider an arbitrary p 2 E .

[Need to show: p2 is even, which meansthere exists integer k such that p2 = 2k .]

Because p 2 E , we know p 2 Z and p is even, and hence there is aninteger ` such that p = 2`. It follows that

p

2 = (2`)2 = 4`2 = 2 · (2`2).

Because ` is an integer, so is 2`2. Thus, there is an integer k (namely 2`2)such that p2 = 2k , and p

2 is even. Because p was arbitrary, the claim istrue.

(CIS 375) Proof Templates (Sets) 22 Sept 2016 3 / 5

Proving Universal Statements Directly

To prove a statement of the form 8x 2 U,P(x):

1 Consider an arbitrary (i.e., nonspecific) object a 2 U.

2 Show that P(a) is true.

Claim: Let E = {n 2 Z : n is even}. For all m 2 E , m2

is even.

Proof: (direct)Consider an arbitrary p 2 E . [Need to show: p2 is even,

which meansthere exists integer k such that p2 = 2k .]

Because p 2 E , we know p 2 Z and p is even, and hence there is aninteger ` such that p = 2`. It follows that

p

2 = (2`)2 = 4`2 = 2 · (2`2).

Because ` is an integer, so is 2`2. Thus, there is an integer k (namely 2`2)such that p2 = 2k , and p

2 is even. Because p was arbitrary, the claim istrue.

(CIS 375) Proof Templates (Sets) 22 Sept 2016 3 / 5

Proving Universal Statements Directly

To prove a statement of the form 8x 2 U,P(x):

1 Consider an arbitrary (i.e., nonspecific) object a 2 U.

2 Show that P(a) is true.

Claim: Let E = {n 2 Z : n is even}. For all m 2 E , m2

is even.

Proof: (direct)Consider an arbitrary p 2 E . [Need to show: p2 is even, which meansthere exists integer k such that p2 = 2k .]

Because p 2 E , we know p 2 Z and p is even, and hence there is aninteger ` such that p = 2`. It follows that

p

2 = (2`)2 = 4`2 = 2 · (2`2).

Because ` is an integer, so is 2`2. Thus, there is an integer k (namely 2`2)such that p2 = 2k , and p

2 is even. Because p was arbitrary, the claim istrue.

(CIS 375) Proof Templates (Sets) 22 Sept 2016 3 / 5

Proving Universal Statements Directly

To prove a statement of the form 8x 2 U,P(x):

1 Consider an arbitrary (i.e., nonspecific) object a 2 U.

2 Show that P(a) is true.

Claim: Let E = {n 2 Z : n is even}. For all m 2 E , m2

is even.

Proof: (direct)Consider an arbitrary p 2 E . [Need to show: p2 is even, which meansthere exists integer k such that p2 = 2k .]

Because p 2 E , we know p 2 Z and p is even, and hence there is aninteger ` such that p = 2`. It follows that

p

2 = (2`)2 = 4`2 = 2 · (2`2).

Because ` is an integer, so is 2`2. Thus, there is an integer k (namely 2`2)such that p2 = 2k , and p

2 is even. Because p was arbitrary, the claim istrue.

(CIS 375) Proof Templates (Sets) 22 Sept 2016 3 / 5

Proofs about Subsets (An Important Universal Statement)

Recall the definition of subset:

W ✓ Z provided that 8x (if x 2 W , then x 2 Z )

To show that W ✓ Z :1 Consider an arbitrary (i.e., nonspecific) a 2 W .

2 Show that a 2 Z .

To show that W 6✓ Z :1 Give a specific object v .

2 Show that v 2 W and v 62 Z .

(CIS 375) Proof Templates (Sets) 22 Sept 2016 4 / 5

Proofs about Subsets (An Important Universal Statement)

Recall the definition of subset:

W ✓ Z provided that 8x (if x 2 W , then x 2 Z )

To show that W ✓ Z :1 Consider an arbitrary (i.e., nonspecific) a 2 W .

2 Show that a 2 Z .

To show that W 6✓ Z :1 Give a specific object v .

2 Show that v 2 W and v 62 Z .

(CIS 375) Proof Templates (Sets) 22 Sept 2016 4 / 5

Proofs about Subsets (An Important Universal Statement)

Recall the definition of subset:

W ✓ Z provided that 8x (if x 2 W , then x 2 Z )

To show that W ✓ Z :1 Consider an arbitrary (i.e., nonspecific) a 2 W .

2 Show that a 2 Z .

To show that W 6✓ Z :1 Give a specific object v .

2 Show that v 2 W and v 62 Z .

(CIS 375) Proof Templates (Sets) 22 Sept 2016 4 / 5

Proofs about Subsets (An Example)

To show that W ✓ Z :1 Consider an arbitrary a 2 W .

2 Show that a 2 Z .

Example

Claim: Let A and B be sets. Then A \ B ✓ A [ B .

Proof: (direct)Consider an arbitrary w 2 A \ B . [NTS: w 2 A [ B .]

By the definition of A \ B , we know w 2 A. Thus, by thedefinition of [, w 2 A [ B .

Because w was an arbitrary element of A \ B ,A \ B ✓ A [ B .

(CIS 375) Proof Templates (Sets) 22 Sept 2016 5 / 5

Proofs about Subsets (An Example)

To show that W ✓ Z :1 Consider an arbitrary a 2 W .

2 Show that a 2 Z .

Example

Claim: Let A and B be sets. Then A \ B ✓ A [ B .

Proof: (direct)Consider an arbitrary w 2 A \ B . [NTS: w 2 A [ B .]

By the definition of A \ B , we know w 2 A. Thus, by thedefinition of [, w 2 A [ B .

Because w was an arbitrary element of A \ B ,A \ B ✓ A [ B .

(CIS 375) Proof Templates (Sets) 22 Sept 2016 5 / 5

Proofs about Subsets (An Example)

To show that W ✓ Z :1 Consider an arbitrary a 2 W .

2 Show that a 2 Z .

Example

Claim: Let A and B be sets. Then A \ B ✓ A [ B .

Proof: (direct)Consider an arbitrary w 2 A \ B .

[NTS: w 2 A [ B .]

By the definition of A \ B , we know w 2 A. Thus, by thedefinition of [, w 2 A [ B .

Because w was an arbitrary element of A \ B ,A \ B ✓ A [ B .

(CIS 375) Proof Templates (Sets) 22 Sept 2016 5 / 5

Proofs about Subsets (An Example)

To show that W ✓ Z :1 Consider an arbitrary a 2 W .

2 Show that a 2 Z .

Example

Claim: Let A and B be sets. Then A \ B ✓ A [ B .

Proof: (direct)Consider an arbitrary w 2 A \ B . [NTS: w 2 A [ B .]

By the definition of A \ B , we know w 2 A. Thus, by thedefinition of [, w 2 A [ B .

Because w was an arbitrary element of A \ B ,A \ B ✓ A [ B .

(CIS 375) Proof Templates (Sets) 22 Sept 2016 5 / 5

Proofs about Subsets (An Example)

To show that W ✓ Z :1 Consider an arbitrary a 2 W .

2 Show that a 2 Z .

Example

Claim: Let A and B be sets. Then A \ B ✓ A [ B .

Proof: (direct)Consider an arbitrary w 2 A \ B . [NTS: w 2 A [ B .]

By the definition of A \ B , we know w 2 A.

Thus, by thedefinition of [, w 2 A [ B .

Because w was an arbitrary element of A \ B ,A \ B ✓ A [ B .

(CIS 375) Proof Templates (Sets) 22 Sept 2016 5 / 5

Proofs about Subsets (An Example)

To show that W ✓ Z :1 Consider an arbitrary a 2 W .

2 Show that a 2 Z .

Example

Claim: Let A and B be sets. Then A \ B ✓ A [ B .

Proof: (direct)Consider an arbitrary w 2 A \ B . [NTS: w 2 A [ B .]

By the definition of A \ B , we know w 2 A. Thus, by thedefinition of [, w 2 A [ B .

Because w was an arbitrary element of A \ B ,A \ B ✓ A [ B .

(CIS 375) Proof Templates (Sets) 22 Sept 2016 5 / 5

Proofs about Subsets (An Example)

To show that W ✓ Z :1 Consider an arbitrary a 2 W .

2 Show that a 2 Z .

Example

Claim: Let A and B be sets. Then A \ B ✓ A [ B .

Proof: (direct)Consider an arbitrary w 2 A \ B . [NTS: w 2 A [ B .]

By the definition of A \ B , we know w 2 A. Thus, by thedefinition of [, w 2 A [ B .

Because w was an arbitrary element of A \ B ,A \ B ✓ A [ B .

(CIS 375) Proof Templates (Sets) 22 Sept 2016 5 / 5